PLEASE HELP
I need this done is 13 minutes.

Polygon ABCD with vertices at A(1,-1) B(3,-1), C(3,-2) and D(1-2) is dilated to create polygon ABCD with vertices at A’(2,-2), B(6,-2), C’(6,-4) and D’(2,-4). Determine the scale factor used to create the image

A. 3
B. 2
C. 1/2
D. 1/3

The length of side X in simple radical form with a rational denominator is 25/√3.

In Mathematics and Geometry, a 30-60-90 triangle is also referred to as a special right-angled triangle and it can be defined as a type of right-angled triangle whose angles are in the ratio 1:2:3 and the side lengths are in the ratio 1:√3:2.

This ultimately implies that, the length of the hypotenuse of a 30-60-90 triangle is double (twice) the length of the shorter leg (adjacent side), and the length of the longer leg (opposite side) of a 30-60-90 triangle is √3 times the length of the shorter leg (adjacent side):

Adjacent side = 5/√3

Hypotenuse, x = 5 × 5/√3

Hypotenuse, x = 25/√3.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

The Equation of circle for Circle B is (x - 2) ² + (y + 4)²  = 38.4.

For circle A:

Radius of circle A = √36 = 6

Area of circle A = π(6)^2 = 36π

Let's assume the radius of circle B is r.

Circumference of circle A = 2π(6) = 12π

Circumference of circle B = 3.2 x Circumference of circle A

= 3.2 x 12π

= 38.4π

Since the ratio of the areas is equal to the ratio of the circumferences, we have:

Area of circle B / Area of circle A = (r^2π) / (36π) = 38.4π / 36π

Simplifying, we get:

r² / 36 = 38.4 / 36

Cross-multiplying, we have:

36 x r² = 38.4 x 36

Dividing both sides by 36:

r^2 = 38.4

Taking the square root of both sides:

r = √38.4

Now we have the radius of circle B. Let's substitute this value into the equation for circle B:

(x - 2) ² + (y + 4)² = (√38.4)²

Simplifying:

(x - 2) ² + (y + 4)²  = 38.4

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Answer:

1335.6

Step-by-step explanation:

volume = pir^2 h = (3.14)(6)^2(15) = (3.14)(36)(15) = (47.1)(36) = 1,335.6

The volume of the tank is 1695.60 ft³.

A cylinder is a three-dimensional object. It is a prism with a circular base. The volume is the amount of space in an object.

Volume of a cylinder = πr²h

Where:

= 3.14 x 6² x 15 = 1695.60 ft³

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The circle should be a circle centered at point (4,-1.5) and have a radius of 2.The equation (x - 4)^2 + (y + 1.5)^2 = 4 represents a circle with a center at (4, -1.5) and a radius of 2.

To identify the graph that shows the position and radius of the wheels, we look for a graph that depicts a circle with a center at (4, -1.5) and a radius of 2.

Among the options provided, the graph that corresponds to the given equation is likely the one that shows a circle with its center at (4, -1.5) and a radius of 2.

The equation provided is a standard form of a circle equation which states that the center of the circle is (4,-1.5) and the radius is 2 (square root of 4).  Any graph with a circle centered at (4,-1.5) and a radius of 2 will represent the position and radius of the wheels of the test model.

Of the options given in the answer section, option (D) is the graph that shows the position and radius of the wheels since it represents a circle with center (4,-1.5) and a radius of 2.

It is also important to note that a circle equation in standard form is (x-h)^2+(y-k)^2=r^2, where (h,k) is the center of the circle and r is the radius.

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Based on the calculations using unit pricing, Toilet paper B has a lower unit price and is the better deal between the two. Customers will save more money if they purchase toilet paper B rather than toilet paper A because the price per mega roll is lower. (option b)

Saving money while shopping is one of the best ways to reduce your expenses and have more disposable income for other things. Unit pricing is a pricing system that displays prices in standard units, allowing customers to compare prices across different brands and package sizes and save money. To determine which toilet paper deal is better, we can use the unit price method, which divides the price by the number of units in the package. The toilet paper with the lower unit price is the better deal.
The formula for unit pricing is as follows:
Unit price = total price ÷ number of units
Using the above formula, we can calculate the unit price of toilet paper A and toilet paper B as follows:
For Toilet paper A:
Unit price = $4.59 ÷ 6 mega rolls
Unit price = $0.765 per mega roll
For Toilet paper B:
Unit price = $9.02 ÷ 12 mega rolls
Unit price = $0.751 per mega roll
Therefore, based on the calculations, Toilet paper B has a lower unit price and is the better deal between the two. Customers will save more money if they purchase toilet paper B rather than toilet paper A because the price per mega roll is lower. (option b)
Unit pricing is a great way to compare prices, and consumers should always use it to determine the better deal between two products, as this will help them save money and stick to their budget.

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It would cost approximately $119.42 to install the molding around the circular room, assuming the cost per foot is $4.22 and the diameter of the room is 9 feet.

To calculate the cost of installing molding around a circular room, we need to find the circumference of the room. The circumference of a circle is given by the formula C = πd, where C is the circumference and d is the diameter.

In this case, the diameter is given as 9. We can substitute this value into the formula to find the circumference:

C = π * 9

C ≈ 28.27 feet

Now that we know the circumference of the room is approximately 28.27 feet, we can calculate the cost of installing the molding. The cost per foot is given as $4.22.

Cost = Cost per foot * Circumference

Cost = $4.22 * 28.27

Cost ≈ $119.42

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The sum of angles ∠D and ∠F in quadrilateral DEFG, inscribed in circle P, is equal to 180∘.

To prove that m∠D + m∠F = 180∘, we can use the property of angles inscribed in a circle.

In a circle, an inscribed angle is equal to half the measure of its intercepted arc. Therefore, if we can show that arc DE + arc FG = 360∘, we can conclude that m∠D + m∠F = 180∘.

Let's start the proof:

1. Quadrilateral DEFG is inscribed in circle P. This means that all the vertices of the quadrilateral lie on the circumference of the circle.

2. Let's consider arc DE and arc FG. These arcs are intercepted by angles ∠D and ∠F, respectively.

3. By the property of angles inscribed in a circle, we know that the measure of an inscribed angle is equal to half the measure of its intercepted arc.

4. Therefore, m∠D = 1/2(arc DE) and m∠F = 1/2(arc FG).

5. We want to prove that m∠D + m∠F = 180∘. This is equivalent to showing that 1/2(arc DE) + 1/2(arc FG) = 180∘.

6. Combining the fractions, we have 1/2(arc DE + arc FG) = 180∘.

7. Now, we need to show that arc DE + arc FG = 360∘.

8. Since quadrilateral DEFG is inscribed in circle P, the sum of the measures of all the arcs intercepted by the sides of the quadrilateral is equal to 360∘.

9. This means that arc DE + arc EF + arc FG + arc GD = 360∘.

10. However, we can observe that arc EF and arc GD are opposite sides of the same chord, so they have equal measures. Therefore, arc EF = arc GD.

11. Substituting arc GD with arc EF in the equation from step 9, we have arc DE + arc EF + arc FG + arc EF = 360∘.

12. Simplifying the equation, we get 2(arc DE + arc EF + arc FG) = 360∘.

13. Dividing both sides by 2, we have arc DE + arc EF + arc FG = 180∘.

14. Comparing this result with step 7, we can conclude that arc DE + arc FG = 180∘.

15. Finally, going back to our initial goal, we can now substitute arc DE + arc FG with 180∘ in the equation from step 6: 1/2(180∘) = 180∘.

16. Simplifying, we have 90∘ = 180∘, which is a true statement.

17. Therefore, we have proven that m∠D + m∠F = 180∘.

Thus, we have successfully proved that the sum of angles ∠D and ∠F in quadrilateral DEFG, inscribed in circle P, is equal to 180∘.

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Angle D measures 81 degrees in quadrilateral ABCD.

We have,

To find the measure of angle D in quadrilateral ABCD, we can use the fact that the sum of the angles in any quadrilateral is always 360 degrees.

Let's denote angle D as x. Given that angle A is 72 degrees, angle B is 94 degrees, and angle C is 113 degrees, we can set up the equation:

72 + 94 + 113 + x = 360

Combining the known angle measures:

279 + x = 360

To solve for x, we can subtract 279 from both sides of the equation:

x = 360 - 279

x = 81

Therefore,

Angle D measures 81 degrees in quadrilateral ABCD.

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The median of the data set are as follows;

Boys: 45. Girls: 110.

The range of the data set are as follows;

Boys: 45. Girls: 110.

The median and range of girls is greater than the median and range of boys.

In Mathematics, a median refers to the middle number (center) of a sorted data set, which is when the data set has either been arranged in a descending order, from the greatest to least or in an ascending order, from the least to greatest.

Based on the information provided in the line plot above, we would determine the median for the data set as follows;

Median of boys = [5th + 6th]/2.

Median of boys = [90 + 90]/2.

Median of boys = 45.

Median of girls = [5th + 6th]/2.

Median of girls = [100 + 120]/2.

Median of girls = 110.

Next, we would determine the range of the data set as follows;

Range = Highest number - Lowest number

Range of boys = 120 - 60

Range of boys = 60.

Range of girls = 120 - 60

Range of girls = 150 - 70.

Range of girls = 80.

In conclusion, we can logically deduce that the median and range for the girls is greater than the median and range of boys.

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It should be noted that the number of sample size based on the information will be 300.

In order to calculate the sample size, we can use the following formula:

n = z² * p * (1-p) / E²

n = 1.96² * 0.5 * (1-0.5) / 0.05²

= 300

A questionnaire is a method of gathering data that makes use of written questions to be answered by the respondents. It is a popular method of data collection because it is relatively easy to administer and can be used to collect a wide variety of information.

In the first column, the population is the entire National Capital Region (NCR). The sample is the city of Manila. In the second column, the population is all STEM students. The sample is all academic track students. In the third column, the population is all the tablespoons of sugar in the jar. The sample is one tablespoon of sugar. In the fourth column, the population is all the vowels in the word "juice". The sample is the vowel "i".

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The percent decrease in total principal and interest paid between a 30-year term mortgage and a 15-year mortgage with a principal balance of $484,500.00 and a 6.5% APR is 31.0%. (Option A)

To calculate the  percent decrease, we can use the following formula.

Percent decrease = ( (30-year factor -   15-year factor) / 30-year factor) x 100

Substituting the values from the chart.  

= (($ 6.65 - $8.71) / $6.65) x 100

= ( -$ 2.06 / $6.65) * 100

= -0.309 *  100

Percent decrease ≈ - 30.9 %

Since the question asks for the percent decrease as a positive value, we take the absolute value of the result which is

Percent decrease ≈ 31%

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The missing part indicated in the figures are ? = 12, TX = 9 and x = 20

Figure a

The missing part can be calculated using the following equation

?/(11 - 5) = 22/11

Evaluate the difference

?/6 = 22/11

So, we have

? = 6 * 22/11

Evaluate the expression

? = 12

Figure b

The missing part can be calculated using the following equation

TX/3 = 6/2

So, we have

TX = 3 * 6/2

Evaluate

TX = 9

Figure c

The value of x can be calculated using the following equation

1/4x + 6 = 2x - 29

So, we have

x + 24 = 8x - 116

Evaluate

-7x = -140

Divide

x = 20

Hence, the value of x is 20

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c) Mode: 15 (appears 6 times)

Mean: 15.25

Median: 15

d) Probability of being 13 years old: 2/20 = 0.1 or 10%

e) Probability of being 14 or 16 years old: 6/20 = 0.3 or 30%

f) Probability of not having 15 years old: 1 - (6/20) = 14/20 = 0.7 or 70%

a) To create a table and calculate the absolute frequency, relative frequency, and percentual frequency, we organize the given ages in ascending order:

13 13 14 14 15 15 15 15 15 16 16 17 17 17 18 18

Age Absolute Frequency Relative Frequency Percentual Frequency

13 2 2/16 12.5%

14 2 2/16 12.5%

15 5 5/16 31.25%

16 2 2/16 12.5%

17 3 3/16 18.75%

18 2 2/16 12.5%

b) To represent the data graphically with a histogram, we plot the ages on the x-axis and the absolute frequency on the y-axis:

c) To calculate the mode, mean, and median:

Mode: The mode is the most frequently occurring age in the dataset. In this case, the mode is 15, as it appears 5 times.

Mean: The mean is the average of all the ages. To calculate it, we sum up all the ages and divide by the total number of ages:

(13 + 13 + 14 + 14 + 15 + 15 + 15 + 15 + 15 + 16 + 16 + 17 + 17 + 17 + 18 + 18) / 20 = 306 / 20 = 15.3

Median: The median is the middle value in the dataset when arranged in ascending order. In this case, the median is 15 since it falls in the middle.

d) To calculate the probability of having 13 years old, we divide the absolute frequency of 13 (2) by the total number of ages (20):

Probability of 13 years old = 2/20 = 0.1 or 10%

e) To calculate the probability of having 14 or 16 years old, we add the absolute frequencies of 14 and 16 (2 + 2 = 4) and divide by the total number of ages (20):

Probability of 14 or 16 years old = 4/20 = 0.2 or 20%

f) To calculate the probability of not having 15 years old, we subtract the absolute frequency of 15 (5) from the total number of ages (20):

Probability of not having 15 years old = (20 - 5)/20 = 0.75 or 75%

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How would you describe the difference between the graphs of f(x) = 2x² and g(x) = -2x²: D. g(x) is a reflection of f(x) over the x-axis.

In Mathematics and Geometry, a reflection over or across the x-axis is represented by this transformation rule (x, y) → (x, -y).

This ultimately implies that, a reflection over or across the x-axis would maintain the same x-coordinate while the sign of the y-coordinate changes from positive to negative or negative to positive.

In this context, we can reasonably infer and logically deduce that a graph of transformed function g(x) = -2x² would be created by applying a reflection over or across the x-axis to the graph of the parent function f(x) = 2x² as shown in the image attached below.

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The total amount of meat bought is 46/4 or 11.5 kg (or 11.5 kilograms).

To express the total amount of meat bought in fraction and decimal form, we need to add up the quantities of each type of meat.

The given quantities are:

2 kg and a quarter of beef

5 kg and two quarters (or a half) of chicken

4 kg and three quarters of pork loin

To find the total amount of meat bought, we can add these quantities:

2 kg and 1/4 + 5 kg and 1/2 + 4 kg and 3/4

To add these mixed numbers and fractions, we need to find a common denominator. The common denominator here is 4.

2 kg and 1/4 can be converted to 9/4 by multiplying 2 by 4 and adding 1.

5 kg and 1/2 can be converted to 9/2 by multiplying 5 by 2 and adding 1.

4 kg and 3/4 remains the same.

Now we can add the fractions:

9/4 + 9/2 + 4 + 3/4

To add the fractions, we need to find a common denominator, which is 4.

9/4 + 9/2 + 16/4 + 3/4

Now we can add the numerators and keep the denominator:

(9 + 18 + 16 + 3) / 4

The numerator becomes 46, so the total amount of meat bought is 46/4.

To express this as a decimal, we can divide the numerator by the denominator:

46 ÷ 4 = 11.5

Therefore, the total amount of meat bought is 46/4 or 11.5 kg (or 11.5 kilograms).

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There are 24 different orders in which Greg, Albert, Joseph, and Finn can take their turns being 'it' in the game of hide-and-seek.

To determine the number of different orders in which the children can take their turns being 'it' in a game of hide-and-seek, we can use the concept of permutations.

Since there are four children (Greg, Albert, Joseph, and Finn), we need to find the number of permutations of these four children. A permutation represents an arrangement of objects in a specific order.

The formula for calculating permutations is given by:

P(n, r) = n! / (n - r)!

Where n is the total number of objects (children) and r is the number of objects (children) to be arranged.

In this case, we have n = 4 (four children) and r = 4 (all four children will take their turns as 'it'). Plugging these values into the formula, we get:

P(4, 4) = 4! / (4 - 4)!

= 4! / 0!

= 4! / 1

= 4 × 3 × 2 × 1 / 1

= 24

Therefore, there are 24 different orders in which Greg, Albert, Joseph, and Finn can take their turns being 'it' in the game of hide-and-seek.

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The volume of the solid obtained by rotating the region enclosed by the curves [tex]y=\sqrt{x^3}[/tex] and [tex]y = x^2[/tex] about the y-axis is [tex]\pi/14[/tex] . option B

To determine the solid's volume after rotating the area bounded by curves

[tex]y = \sqrt{x^3}[/tex] and [tex]y = x^2[/tex]

We can apply the cylindrical shell approach to the y-axis.

First, let's locate the spots where the two curves intersect:

[tex]\sqrt{x^3} = x^2[/tex]

Squaring both sides:

[tex]x^3 = x^4[/tex]

Rearranging:

[tex]x^4 - x^3 = 0[/tex]

Factorizing [tex]x^3(x - 1) = 0[/tex]

Therefore, the locations of intersection are x = 0 and x = 1.

The integral for the solid's volume must then be set up using cylindrical shells. The following formula determines the volume of a cylindrical shell:

V is equal to 2[a, b] x * h(x). dx where a and b are the integration limits, x is the shell's radius, and h(x) is the height of the shell.

The radius of the shell in this instance is x, while the height of the shell is the ratio of the two curves: [tex]h(x) = \sqrt{x^3} - x^2.[/tex]

Since those are the locations of intersection, the range of integration's bounds is 0 to 1.

The integral then becomes:

∫[tex]V = 2\pi [0, 1] x * (\sqrt{x^3} - x^2) dx[/tex]

We can simplify the formula inside the integral in order to evaluate it:

V = 2π ∫[0, 1] (x^(5/2) - x^3) dx

We can integrate each word by applying the power rule for integration as follows:

[tex]V = 2\pi [(2/7)x^{(7/2)} - (1/4)x^{4}] |[0, 1][/tex]

Calculating the limits of the definite integral:

[tex]V = 2\pi [(2/7)(1^{(7/2))} - (1/4)(1^{4)}] - 2\pi [(2/7)(0^{(7/2})) - (1/4)(0^{4)}][/tex]

Simplifying further:

V = 2π [(2/7) - (1/4)]

V = 2π (8/28 - 7/28)

V = 2π/28

Simplifying the fraction:

V = π/14 ,Therefore, the correct answer is option B.

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Answer:

-----------------

Find the differences in the sequence  -11, -7, -3, 1, ...

The difference is common, so the sequence is an AP.

The nth term is:

Find the 52nd term:

Find the 52nd term using the given formula:

Find the previous term:

Find the common difference:

Answer:

[tex]\begin{aligned}\textsf{34)} \quad d&=4\\a_n&=4n-15\\a_{52}&=193\end{aligned}[/tex]

[tex]\begin{aligned}\textsf{35)} \quad d&=-4\\a_{52}&=-238\end{aligned}[/tex]

Step-by-step explanation:

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.

Given sequence:

To determine if the given sequence is arithmetic, calculate the differences between consecutive terms.

[tex]a_4-a_3=1-(-3)=4[/tex]

[tex]a_3-a_2=-3-(-7)=4[/tex]

[tex]a_2-a_1=-7-(-11)=4[/tex]

As the differences are constant, the sequence is arithmetic, with common difference, d = 4.

The explicit formula for an arithmetic sequence is:

[tex]\boxed{a_n=a+(n-1)d}[/tex]

where:

To find the explicit formula for the given sequence, substitute a = -11 and d = 4 into the formula:

[tex]\begin{aligned}a_n&=-11+(n-1)4\\&=-11+4n-4\\&=4n-15\end{aligned}[/tex]

To find the 52nd term, simply substitute n = 52 into the formula:

[tex]\begin{aligned}a_{52}&=4(52)-15\\&=208-15\\&=193\end{aligned}[/tex]

Therefore, the 52nd term is a₅₂ = 193.

[tex]\hrulefill[/tex]

Given explicit formula for an arithmetic sequence:

[tex]a_n=-30-4n[/tex]

To find the common difference, we need to compare it with the explicit formula for the nth term:

[tex]\begin{aligned}a_n&=a+(n-1)d\\&=a+dn-d\\&=a-d+dn\end{aligned}[/tex]

The coefficient of the n-term is -4, therefore, the common difference is d = -4.

To find the 52nd term, simply substitute n = 52 into the formula:

[tex]\begin{aligned}a_{52}&=-30-4(52)\\&=-30-208\\&=-238\end{aligned}[/tex]

Therefore, the 52nd term is a₅₂ = -238.

The coordinates of the vertices for the image of the pre image is translated 4 units right are:

C. A' = (-1,2), B' = (5, 3), C' = (1,0)

A geometric transformation involves taking a geometric object as input and producing a new geometric object as output. We have different types of geometric transformations such as translation, rotation, reflection etc.

A translation moves a geometric object a certain distance in a certain direction. Translation to the right means movement in the positive x-direction.

Thus, the coordinates of the vertices for the image of the pre image if it is translated 4 units right can be obtained by adding 4 to x-coordinate values while y-coordinate values remain the same. That is:

A(-5,2) becomes A'(-5+4, 2) = (-1,2)

B(1,3)becomes  B'(1+4, 3) = (5, 3)

C(-3,0) becomes C'(-3+4, 0) = (1,0)

Therefore, the coordinates of the vertices of the image of triangle ABC after it is translated 4 units right are (-1, 2), (5, 3), and (1, 0).

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We subtract the cost of wine when buying one bottle per week from the cost of wine when buying two bottles per weekend. Miranda saves $780 in one year by buying only one bottle of wine per week instead of two.

To calculate how much Miranda saves in one year by buying only one bottle of wine per week instead of two, we can follow these steps:

Determine the number of bottles of wine she buys in a year:

Miranda used to buy 2 bottles of wine per weekend, so in a week, she bought 2 bottles.

Since there are 52 weeks in a year, the number of bottles she bought in a year is 2 bottles/week * 52 weeks = 104 bottles.

Calculate the total cost of wine for the year:

Since each bottle costs $15, the total cost of wine for the year when buying 2 bottles per week would be 104 bottles * $15/bottle = $1560.

Calculate the cost of wine for one year when buying only one bottle per week:

With Miranda's decision to buy one bottle per week, the total number of bottles she buys in a year is 1 bottle/week * 52 weeks = 52 bottles.

Therefore, the cost of wine for one year when buying only one bottle per week would be 52 bottles * $15/bottle = $780.

Calculate the amount saved in one year:

To determine the amount saved, we subtract the cost of wine when buying one bottle per week from the cost of wine when buying two bottles per weekend.

Amount saved = $1560 - $780 = $780.

Therefore, Miranda saves $780 in one year by buying only one bottle of wine per week instead of two.

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The Coordinates of the terminal point determined by `t = 1177` are approximately `(-0.17101, -0.98526)`.Thus, the correct option is B (-0.17101, -0.98526).

The following trigonometric function represents a point P on the unit circle as a function of the angle θ in standard position: `(cos θ, sin θ)`.

Here, we have to determine the coordinates of the terminal point determined by `t = 1177`.Terminal Point on a unit circle:The terminal point is a point on a circle that lies on the terminal side of an angle. The angles that end on the same terminal side of the x-axis are called coterminal angles. The angle between the positive x-axis and the line segment connecting the origin of the circle and the point on the circle is called the angle in the standard position.

So, the angle measure `t = 1177` may be too large or too small to locate the corresponding point P on the unit circle. To find an equivalent angle between 0 and 360 degrees, we may subtract or add an integral number of revolutions: `1177 - 360 × 3 = 97`.That is, an angle of `t = 1177` degrees is coterminal with an angle of `t = 97` degrees. The terminal point determined by `t = 1177` is the same as the terminal point determined by `t = 97`.

The point on the unit circle with `t = 97` degrees lies in the fourth quadrant because the standard angle of 97 degrees is obtained by rotating a ray 97 degrees counterclockwise from the positive x-axis.So, `P = (cos 97°, sin 97°)`.We can approximate the values of `cos 97°` and `sin 97°` using a calculator or computer software as: `cos 97° ≈ -0.17101` and `sin 97° ≈ -0.98526`.

Therefore, the coordinates of the terminal point determined by `t = 1177` are approximately `(-0.17101, -0.98526)`.Thus, the correct option is B (-0.17101, -0.98526).

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first off, let's take a peek at the picture above

hmmm the hyperbola is opening sideways, that means it has a horizontal traverse axis, it also means that the positive fraction will be the one with the "x" variable in it.

now, the length of the horizontal traverse axis is 4 units, from vertex to vertex, that means the "a" component of the hyperbola is half that or 2 units, and 2² = 4, with a center at the origin.

[tex]\textit{hyperbolas, horizontal traverse axis } \\\\ \cfrac{(x- h)^2}{ a^2}-\cfrac{(y- k)^2}{ b^2}=1 \qquad \begin{cases} center\ ( h, k)\\ vertices\ ( h\pm a, k)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad \sqrt{ a ^2 + b ^2} \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{(x- 0)^2}{ 2^2}-\cfrac{(y- 0)^2}{ (\sqrt{3})^2}=1\implies {\Large \begin{array}{llll} \cfrac{x^2}{4}-\cfrac{y^2}{3}=1 \end{array}}[/tex]

The integral is ln|x + cos(x)| + C, where C represents the constant of integration.

To find the integral of the function 1/(x + cos(x)), we can employ a combination of algebraic manipulation and the use of standard integration techniques. Here's the solution:

First, let's rewrite the integral in a slightly different form to simplify the process:

∫(1/(x + cos(x))) dx

We notice that the denominator, x + cos(x), is not amenable to direct integration. To overcome this, we employ a substitution. Let's set u = x + cos(x). Now, differentiate u with respect to x: du/dx = 1 - sin(x).

Rearranging this equation, we get dx = du/(1 - sin(x)).

Substituting these values, the integral becomes:

∫(1/(u(1 - sin(x)))) du

Next, we simplify further by factoring out 1/(1 - sin(x)) from the integral:

∫(1/(u(1 - sin(x)))) du = ∫(1/u) du = ln|u| + C

Replacing u with its original expression, we have:

ln|x + cos(x)| + C

Therefore, the answer to the integral is ln|x + cos(x)| + C, where C represents the constant of integration.

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The value of [tex]\(r\)[/tex] is [tex]\(r = -2\)[/tex], according to the given cartesian points.

To find the value of [tex]\(r\)[/tex], we can use the slope-intercept form of a linear equation, [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope of the line. Given that the line has a slope of [tex]\(-10\)[/tex] and passes through the cartesian points [tex]\((4, 8)\)[/tex]and \[tex]((5, r)\)[/tex], we can calculate the slope as follows:

[tex]\[m = \frac{{y_2 - y_1}}{{x_2 - x_1}} = \frac{{r - 8}}{{5 - 4}} = r - 8\][/tex]

Since the slope is [tex]\(-10\)[/tex], we can equate it to the calculated slope and solve for [tex]\(r\)[/tex]:

[tex]\[-10 = r - 8\][/tex]

Simplifying the equation, we have:

[tex]\[r - 8 = -10\][/tex]

Adding [tex]\(8\)[/tex] to both sides, we get:

[tex]\[r = -10 + 8\][/tex]

Therefore, the value of [tex]\(r\)[/tex] is [tex]\(r = -2\)[/tex].

In conclusion, the value of [tex]\(r\)[/tex] in the line with a slope of [tex]-10[/tex] passing through the points [tex](4, 8)[/tex] and [tex](5, \(r\))[/tex] is [tex]\(r = -2\)[/tex]. This satisfies the equation and represents the y-coordinate of the second point. This value of [tex]r[/tex] indicates that the second point lies on the line with a slope of -[tex]10[/tex] passing through ([tex]4,8[/tex]).

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The algebra 2 is a vast field that deals with various mathematical concepts, such as equations and inequalities.

Algebra 2 is a branch of mathematics that mainly deals with equations and inequalities. An equation is a mathematical statement that implies that two expressions are equal.

Similarly, an inequality implies that two expressions are not equal but are related by a certain operation.There are different types of equations that one may encounter in Algebra 2.

A linear equation is an equation that can be represented by a straight line on the coordinate plane. A quadratic equation is one that can be written in the form ax² + bx + c = 0.

There are also exponential, logarithmic, and trigonometric equations that one may come across.Inequalities are statements that two expressions are not equal. Inequalities can be represented graphically on a coordinate plane, just like equations.

There are different types of inequalities such as linear, quadratic, exponential, and logarithmic inequalities.In conclusion,  These concepts help in solving real-world problems by providing a framework to analyze them mathematically.

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We can determine the shortest distance from the surface xy+15x+z^2=209 to the origin.

Given the equation of the surface is xy + 15x + z^2 = 209.

Let's determine the shortest distance from the surface to the origin.

The shortest distance between the surface and the origin is given by the perpendicular distance, which can be calculated as follows:

Firstly, we need to determine the gradient of the surface, which is the vector normal to the surface.

For this purpose, we need to write the surface equation in the standard form, which is:xy + 15x + z^2 = 209 xy + 15x + (0)z^2 - 209 = 0 (the coefficients of x, y, and z are a, b, and c).

The gradient of the surface is given by the vector: ∇f = (a, b, c) = (y + 15, x, 2z) at the point P(x, y, z), which is a point on the surface.

Here, the normal vector is ∇f = (y + 15, x, 2z).

Now, let's consider a point A on the surface which is closest to the origin.

Let the coordinates of A be (a, b, c).

Therefore, the position vector of A is given by: OA = ai + bj + ck.

The direction of the position vector is in the direction of the normal vector, and therefore: OA is parallel to ∇f.

Thus, we can write: OA = λ∇f = λ(y + 15)i + λxj + 2λzkWhere λ is a scalar.

Since A lies on the surface, we have: a*b + 15a + c^2 = 209.

We also know that OA passes through the origin.

Therefore, the position vector of A is perpendicular to the direction vector OA.

This gives us: OA·OA = 0⟹ (ai + bj + ck)·(λ(y + 15)i + λxj + 2λzk) = 0

Simplifying this equation gives us:aλ(y + 15) + bλx + c(2λz) = 0

Also, we know that OA passes through the origin.

Therefore, the magnitude of OA is equal to the distance of A from the origin.

Hence, we can write: |OA| = √(a^2 + b^2 + c^2)

The value of λ can be obtained from the equation: aλ(y + 15) + bλx + c(2λz) = 0orλ = -2cz / (b + a(y + 15))

Substituting this value of λ in OA, we get: OA = λ(y + 15)i + λxj + 2λzk= -2cz/(b + a(y + 15)) (y + 15)i - 2cz/(b + a(y + 15)) xj - 4cz^2/(b + a(y + 15))k

Substituting this value of λ in |OA|, we get: |OA| = √[(2cz/(b + a(y + 15)))^2 + (2cz/(b + a(y + 15)))^2 + (4cz^2/(b + a(y + 15)))^2] = 2cz√[(y + 15)^2 + x^2 + 4z^2] / |b + a(y + 15)|

The distance of the point A from the origin is |OA|, which is minimized when the denominator is maximized. The denominator is given by |b + a(y + 15)|.

Thus, we have to maximize the denominator with respect to a and b. The condition for maximum value of the denominator is obtained by differentiating the denominator with respect to a and b separately and equating it to zero. The values of a and b obtained from these equations are substituted in the equation a*b + 15a + c^2 = 209 to obtain the coordinates of the point A, which is closest to the origin.

Hence, we can determine the shortest distance from the surface xy+15x+z^2=209 to the origin.

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The graph should visually represent the shaded regions A and B, as well as their overlap in region AB. It's important to note that without a specific scale or values, the graph may not accurately depict the exact shape and position of the shaded regions.

To graph the system of inequalities 2x - y < 4 and x + y > -1, we can start by graphing each inequality separately and then determining the overlapping region.

For the inequality 2x - y < 4:

Start by graphing the line 2x - y = 4. Choose two points on the line, such as (0, -4) and (2, 0), and connect them to draw the line.

Since the inequality is "less than" (<), shade the region below the line. Label this shaded region as A.

For the inequality x + y > -1:

Graph the line x + y = -1 using points such as (-2, 1) and (0, -1). Connect the points to draw the line.

Since the inequality is "greater than" (>), shade the region above the line. Label this shaded region as B.

Finally, identify the overlapping region of the shaded regions A and B. This region represents the solution to the system of inequalities and is labeled as AB. This region indicates the values of x and y that satisfy both inequalities simultaneously.

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Answer:

Step-by-step explanation:

2x − y < 4

x + y > −1

In each graph, the area for f(x) is shaded and labeled A, the area for g(x) is shaded and labeled B, and the area where they have shading in common is labeled AB.

The correct statement is that they have different x- and y-intercepts, but we cannot determine their end behavior based on the given information.

Based on the information provided, we can compare the two functions f and g as follows:

Function f:

- It passes through the points (-10, -1) and (2, 8).

- It intercepts the x-axis at -2 units and the y-axis at 2 units.

Function g:

- It is represented by the table with x-values -2, -1, 0, 1, 2, and corresponding y-values 0, 2, 8, 26.

Comparing the two functions based on their intercepts and end behavior:

A. They have the same y-intercept and the same end behavior as x approaches ∞: This statement is incorrect because the y-intercepts of f and g are different. Function f intercepts the y-axis at 2 units, while function g intercepts the y-axis at 0 units. Additionally, we do not have information about the end behavior of either function.

B. They have the same x-intercept and the same end behavior as x approaches ∞: This statement is incorrect because the x-intercepts of f and g are different. Function f intercepts the x-axis at -2 units, while function g does not intercept the x-axis.

C. They have different x- and y-intercepts but the same end behavior as x approaches ∞: This statement is partially correct. Function f and g have different x- and y-intercepts, but we don't have information about their end behavior.

D. They have the same x- and y-intercepts: This statement is incorrect as the x- and y-intercepts of f and g are different.

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The only system of inequalities that has no solution is x<2 and x>7.(option-b)

The system of inequalities that has no solution is x<2 and x>7.

If x is less than 2, it cannot be greater than 7 at the same time. These two inequalities are contradictory, and there is no number that could satisfy both of them simultaneously.

The system of inequalities x>2 and x<7 forms an open interval between 2 and 7. Any value of x within this interval can satisfy both inequalities, so this system has a solution.

The system of inequalities x<2 and x<7 forms an inequality that is satisfied by any value of x that is less than 7, so this system also has a solution.

Lastly, the system of inequalities x>2 and x>7 forms an inequality that is not satisfied by any value of x, as there is no number that is simultaneously greater than 2 and greater than 7.(option-b)

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The type of transformation shown is vertical translation.

Since, Transformation of geometrical figures or points is the manipulation of a given figure to some other way.

Different types of transformations are Rotation, Reflection, Glide reflection, Translation and Dilation.

Given a polygon EFGH.

It is transformed to another polygon with the same size E'F'G'H'.

Here the polygon EFGH is just moved downwards as it is and mark is as E'F'G'H'.

If it is rotation or reflection, the points will change it's correspondent place.

Translation is a type of transformation where the original figure is shifted from a place to another place without affecting it's size.

Therefore, here translation is done.

Since the shifting is done vertically, it is vertical translation.

Hence the transformation is vertical translation.

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