The length of BD is approximately option (D) 9.75 units
Ptolemy's theorem is a theorem in Euclidean geometry that relates the sides and diagonals of a cyclic quadrilateral. A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle.
We can use Ptolemy's theorem to find the length of BD
AC × BD = AB × CD + BC × AD
Substituting the given values:
12 × BD = 9 × 7 + 6 × 8
Multiply the numbers
12 × BD = 63 + 48
12 × BD = 111
BD = 111/12
Divide the numbers
BD ≈ 9.75 units
Therefore, the correct option is (D) 9.75 units
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If the sphere shown above has a radius of 10 units, then what is the approximate volume of the sphere?
Answer:
V = 4188.97
Step-by-step explanation:
Formula for volume of a sphere is 4/3(pi)(r^3)
Using the formula for volume of a sphere, we plug in (4/3 * pi * 10^3).
Let a, b E N, with a > 0 and b > 0. (a) Let p, q, r, s be the unique integers such thata = qb + r,0 < r < b,b = pr+s,0 < s
The Euclidean algorithm gives gcd(a,b) = gcd(b,r) and gcd(b,r) = gcd(r,s).
The Euclidean algorithm is a recursive algorithm to find the greatest common divisor (gcd) of two integers a and b. It works by repeatedly finding the remainder of the division of the larger number by the smaller number, until the remainder is 0, at which point the gcd is the last non-zero remainder.
When applying the algorithm to a and b with a > b, we can write a = qb + r, where q is the quotient and r is the remainder. Then, we apply the algorithm to b and r to find the gcd(b,r), and so on until we reach 0. At each step, the gcd of the current pair of numbers is equal to the gcd of the previous pair, since any common divisor of the current pair must also divide the previous pair. Therefore, we have gcd(a,b) = gcd(b,r) = gcd(r,s), where s is the final non-zero remainder.
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1) Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative.)
f(x) = 1/2 + 5/6^x2 − 4/5^x3
The most general antiderivative of f(x) = 1/2 + 5/6x² − 4/5x³ is F(x) = 1/2x + 5/18x³ − 1/5x⁴ + C, where C is the constant of the antiderivative.
To check this answer, we can differentiate F(x) and see if it gives us back f(x). Taking the derivative of F(x), we get f(x) = d/dx (1/2x + 5/18x³ − 1/5x⁴ + C) = 1/2 + 5/6x² − 4/5x³, which matches the original function f(x). Therefore, F(x) is the most general antiderivative of f(x).
The constant of integration, denoted by C, is added because when taking the derivative of a constant, it is equal to zero. Thus, the constant of integration can be any real number, and it is included in the antiderivative to account for all possible functions that have f(x) as their derivative.
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write a function file [a, b, i] gemetry ( across section, area, orientation) that calculates the perimeter of a beam given the desired cross section
This would calculate the perimeter of a square beam with an area of 25 square units, oriented horizontally.
```
function [perimeter] = geometry(across_section, area, orientation)
% Calculate the perimeter of a beam given the desired cross section
% Define constants for the shape of the cross section
switch across_section
case 'square'
side_length = sqrt(area);
num_sides = 4;
case 'circle'
radius = sqrt(area / pi);
num_sides = 0; % Circles have no sides
case 'rectangle'
aspect_ratio = 2; % Set this to whatever you need for your application
width = sqrt(area / aspect_ratio);
height = width * aspect_ratio;
num_sides = 4;
otherwise
error('Unknown cross section type');
end
% Calculate the perimeter based on the number of sides and shape
switch across_section
case 'circle'
perimeter = 2 * pi * radius;
otherwise
perimeter = num_sides * (width + height);
end
% Adjust the perimeter based on the orientation
switch orientation
case 'horizontal'
% No adjustment necessary
case 'vertical'
% Swap the width and height
temp = width;
width = height;
height = temp;
otherwise
error('Unknown orientation type');
end
end
```
To use this function, you would call it with the desired values for `across_section`, `area`, and `orientation`. For example:
```
perimeter = geometry('square', 25, 'horizontal');
```
This would calculate the perimeter of a square beam with an area of 25 square units, oriented horizontally.
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Area = ?
As in the picture attached.
The area of the shaded area is 0.314m²
How did we reach this conclusion?First let's analyze the shape.
It comprises:
A square whose side length is 1m
Then there is the circle that passes through a corner of the square and is tangent to the opposite two sides of the square. So the region shaded in blue is the region contained by the circle and the square but not common to both shapes.
To get the area of this portion shaded in blue, here is what we must do:
So the Area of the region shaded blue is:
Area of the circle + area of the square - 2 x Area of the overlapping area.
To further help our analysis, we must deconstruct this shape into familiar ones by creating a composite right triangle as shown in the attached image.
Since we have a right triangle, the arc that it subtends will be a 180° arc thus, the hypotenuse of the triangle is the diameter of the circle.
Thus, when we created a right triangle from the shape, we also carved out a semi-circle.
This means that the overlapping areas are:
Area of the composite right triangle + the Area of the Semi Circle.
Since we now have two semi circles, we can cancel out the area of the circle above.
Our new equation for the area shaded in blue becomes:
The area of the square - the area of 2 Right Triangles
Now we can being to compute for the area of the shapes:
Taking the right triangle firs, create a line from the radius bisecting the triangle at 90 degrees. This gives us two equal smaller right triaangles.
Since the ∠90 is bisected, this means that the sides will be r√2 and the height of both right triangles = r (radius)
extending the radius into the opposite direcitn, we now have the diagonal of the square.
Diagonal = r + r√2 = √2
Solving for r we say:
r(1+√2) = √2
r = √2/(1+√2)
To further simplify the above, we can multiply it by the conjugate:
r = √2/(1+√2) x (1-√2)/(1-√2)
Thus,
r = 2-√2
So since the length of the square is 1m, then the area = 1x1 = 1m
The area of the composite triangle =( r√2 x r√2)/ 2 = r²
Since the area of the shaded region as given above is
The area of the square - the area of 2 Right Triangles
Then it's area = 1 - 2r²
We can simplify this further to get:
1-2(2-√2)²
= 8 √2 - 11m²
area of the shaded region ≈ 0.314m²
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6
Write the sum in expanded form. ∑ = 23 / (i + 23)
i=1
The sum in expanded form is 23 / 24 + 23 / 25 + 23 / 26 + 23 / 27 + 23 / 28 + 23 / 29.
The sum in expanded form is given by the expression 23 / (i + 23), where i varies from 1 to 6.
The sum in expanded form can be calculated by substituting the values of i from 1 to 6 into the expression 23 / (i + 23) and summing them up.
When i = 1, the expression becomes 23 / (1 + 23) = 23 / 24.
When i = 2, the expression becomes 23 / (2 + 23) = 23 / 25.
When i = 3, the expression becomes 23 / (3 + 23) = 23 / 26.
When i = 4, the expression becomes 23 / (4 + 23) = 23 / 27.
When i = 5, the expression becomes 23 / (5 + 23) = 23 / 28.
When i = 6, the expression becomes 23 / (6 + 23) = 23 / 29.
Therefore, the sum in expanded form is 23 / 24 + 23 / 25 + 23 / 26 + 23 / 27 + 23 / 28 + 23 / 29.
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The sum in expanded form is 23 / 24 + 23 / 25 + 23 / 26 + 23 / 27 + 23 / 28 + 23 / 29.
The sum in expanded form is given by the expression 23 / (i + 23), where i varies from 1 to 6.
The sum in expanded form can be calculated by substituting the values of i from 1 to 6 into the expression 23 / (i + 23) and summing them up.
When i = 1, the expression becomes 23 / (1 + 23) = 23 / 24.
When i = 2, the expression becomes 23 / (2 + 23) = 23 / 25.
When i = 3, the expression becomes 23 / (3 + 23) = 23 / 26.
When i = 4, the expression becomes 23 / (4 + 23) = 23 / 27.
When i = 5, the expression becomes 23 / (5 + 23) = 23 / 28.
When i = 6, the expression becomes 23 / (6 + 23) = 23 / 29.
Therefore, the sum in expanded form is 23 / 24 + 23 / 25 + 23 / 26 + 23 / 27 + 23 / 28 + 23 / 29.
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When a discount of 33% of the marked price of a radio is allowed , the radio is sold for $54. How much discount does Raymond get when buying the radio?
Raymond gets a discount of $27.
Let the marked price of the radio be 'x'.
According to the problem, a discount of 33% is given, which means that the selling price is 67% of the marked price.
So, the selling price of the radio is 67% of x, which is given as $54 in the problem.
Hence, 67% of x = $54
Solving for x, we get x = $80
The discount amount is the difference between the marked price and selling price, which is $80 - $54 = $26.
Therefore, Raymond gets a discount of $26.
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Which of the following fractions are equal to 3/4?
Answer:
6/8, 9/12, 12/16, 15/20
Step-by-step explanation:
If you simplify the fractions it will all equal 3/4.
Simplify by finding the Greatest Common Factor and dividing both numbers by the GCF.
a federal report indicated that 17 % of children under age 6 live in poverty in washington, an increase over previous years. how large a sample is needed to estimate the true proportion of children under age living in poverty in washington within with confidence? round the intermediate calculations to three decimal places and round up your final answer to the next whole number.
We would need a sample size of at least 1073 children under age 6 in Washington to estimate the true proportion of children living in poverty with 95% confidence and a margin of error of 2%.
To estimate the true proportion of children under age 6 living in poverty in Washington with 95% confidence and a margin of error of 2%, we can use the formula:
n = (Z² * p * q) / E²
where:
Z = the Z-score corresponding to the desired confidence level (1.96 for 95% confidence)
p = the estimated proportion (0.17 based on the federal report)
q = 1 - p
E = the desired margin of error (0.02)
Plugging in these values, we get:
n = (1.96² * 0.17 * 0.83) / 0.02²
n = 1072.45
Rounding up to the next whole number, we would need a sample size of at least 1073 children under age 6 in Washington to estimate the true proportion of children living in poverty with 95% confidence and a margin of error of 2%.
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Todd spent some time cleaning his room
Jeff spent 11 more minutes cleaning his room room than tod spent.Jeff spent 43
minutes
Answer:
32
Step-by-step explanation:
43-11=32
. find a set of smallest possible size that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets
Hi! To find a set of the smallest possible size that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets, follow these steps:
1. List out the given subsets: {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}.
2. Combine the elements of both subsets without repeating any numbers: {1, 2, 3, 4, 5, 6, 8, 10}.
3. The combined set is {1, 2, 3, 4, 5, 6, 8, 10}, which has a size of 8.
So, the smallest possible set that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets is {1, 2, 3, 4, 5, 6, 8, 10}.
The smallest possible set that includes both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} subsets is {1, 2, 3, 4, 5, 6, 8, 10}.
To find a set of the smallest possible size that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets, follow these steps:
1. Identify the given subsets: {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}.
2. Combine the elements from both subsets without repeating any numbers.
3. Organize the combined elements in ascending order.
Your answer: The smallest possible set that includes both subsets is {1, 2, 3, 4, 5, 6, 8, 10}.
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Hi! To find a set of the smallest possible size that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets, follow these steps:
1. List out the given subsets: {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}.
2. Combine the elements of both subsets without repeating any numbers: {1, 2, 3, 4, 5, 6, 8, 10}.
3. The combined set is {1, 2, 3, 4, 5, 6, 8, 10}, which has a size of 8.
So, the smallest possible set that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets is {1, 2, 3, 4, 5, 6, 8, 10}.
The smallest possible set that includes both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} subsets is {1, 2, 3, 4, 5, 6, 8, 10}.
To find a set of the smallest possible size that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets, follow these steps:
1. Identify the given subsets: {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}.
2. Combine the elements from both subsets without repeating any numbers.
3. Organize the combined elements in ascending order.
Your answer: The smallest possible set that includes both subsets is {1, 2, 3, 4, 5, 6, 8, 10}.
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PLEASE HELP!!
20. The table below shows the cost of flying from San Francisco to various other cities in the United States. There is a relationship between the distance you are flying and the cost of your plan ticket. The data from the table is represented on the scatter plot. Distance (miles) Cost of the plane ticket ($) 300 Cost of 250 plane ticket ($) 200 150 100 0 374 1,240 200 600 143 125 250 725 150 110 180 1,100 950 1,500 224 180 250 500 750 1000 1250 Distance (miles) 500 164 A) Draw a line of best fit and pick two good points from the table that are on your line B) Determine the equation for the line of best fit.
Having drawn and attached the scatter plot for the given data, the equation for the line of best fit is y = 0.083x + 97.55
How did we arrive at the above conclusion?
After plotting the scatter point, we choose two points on the line of best fit that are far apart.
The points chosen are:
(150, 110) And (1240, 200)
Using the slope-intercept form we proceed to find the equation
y = mx + b
y = cost of plane tickets
x = distance
m = slope
m = (y2-y1)/(x2-x1)
m = (200-110) / (1240 - 150)
m = 0.083
So, we can state
y = mx + b
110 = 0.083 * 150 + b
b = 97.55
hence,
the line of best fit is:
y = 0.083x + 97.55
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1. An online store sells sportswear. Of all online sales, it is known that the amount of each sale is right skewed with mean $55 and standard deviation $19. A sample of 50 sales is randomly selected.(a). Find the mean of the sampling distribution of the mean amount spent per sale for samples of size 50.(b).Find the standard deviation of the sampling distribution from part (a). (Round your answer to three decimal places.)
The following parts can be answered by the concept of standard deviation.
a. The population mean is $55.
b. The standard deviation of the sampling distribution is approximately 2.688 (rounded to three decimal places).
(a) The mean of the sampling distribution of the mean amount spent per sale for samples of size 50 is equal to the population mean. In this case, the population mean is $55.
(b) To find the standard deviation of the sampling distribution, we use the formula:
Standard deviation of the sampling distribution = (Population standard deviation) / sqrt(sample size)
In this case, the population standard deviation is $19 and the sample size is 50. Plugging these values into the formula, we get:
Standard deviation of the sampling distribution = 19 / sqrt(50) ≈ 19 / 7.071 ≈ 2.688
So, the standard deviation of the sampling distribution is approximately 2.688 (rounded to three decimal places).
Therefore,
a. The population mean is $55.
b. The standard deviation of the sampling distribution is approximately 2.688 (rounded to three decimal places).
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The person filling the tank realizes something is wrong with the hose. After 30 minutes
he shuts off the hose and tries a different hose. The second hose flows at a constante
of 18 gallons per minute.
How long does it take to completely fill the tank by using the second hose?
If the person filling the tank realizes something is wrong with the hose. The time it take to completely fill the tank by using the second hose is: 5.56 minutes.
How to find the time?Using this formula find long does it take to completely fill the tank by using the second hose
Let Assume the tank has a capacity of 100 gallons.
Time =Amount of water ÷ rate
Let plug in the formula
Time = 100 gallons ÷ 18 gallons/minute
Time = 5.56 minutes (Approximately)
Therefore it would take 5.56 minutes to fill a 100-gallon tank.
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If the person filling the tank realizes something is wrong with the hose. The time it take to completely fill the tank by using the second hose is: 5.56 minutes.
How to find the time?Using this formula find long does it take to completely fill the tank by using the second hose
Let Assume the tank has a capacity of 100 gallons.
Time =Amount of water ÷ rate
Let plug in the formula
Time = 100 gallons ÷ 18 gallons/minute
Time = 5.56 minutes (Approximately)
Therefore it would take 5.56 minutes to fill a 100-gallon tank.
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Determine the components of the compound statement. (Select all that apply.) 7 + 1 2 7 □7,1,7 □ 7+1 □74157 □7+1=7 □27
The compound statement is not entirely clear as it contains several separate expressions. However, I can break down and analyze each of the parts:
7 + 1 2 7: This is a mathematical expression that represents the sum of 7, 1, and 27, which evaluates to 35.
7,1,7: This is a list of three individual numbers.
7+1: This is a mathematical expression that represents the sum of 7 and 1, which evaluates to 8.
74157: This is a five-digit number with no apparent mathematical relationship to the other expressions.
7+1=7: This is a mathematical equation that tests the equality of the expressions 7+1 and 7. Since 7+1 is not equal to 7, this equation is false.
27: This is a single number that is not obviously related to the other expressions.
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Consider the following hypothesis problem. n = 30 s2 = 625 H0: σ2 =500 Ha:σ2≠500 The test statistic equals a. .63. b. 12.68. c. 13.33. d. 13.68.
The required ‘test statistic’ is 36.25
To solve this problem, we'll use the Chi-squared test statistic for testing the variance of a population. Here are the steps:
Identify the given information:
- Sample size (n) = 30
- Sample variance (s²) = 625
- Null hypothesis (H₀): σ² = 500
- Alternative hypothesis (Hₐ): σ² ≠ 500
Calculate the degrees of freedom (df) using the formula: df = n - 1
- df = 30 - 1 = 29
Calculate the Chi-squared test statistic (χ²) using the formula: χ² = (n - 1) * (s² / σ²)
- χ² = (29) * (625 / 500)
Compute the test statistic value:
- χ² = 29 * (1.25) = 36.25
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This is exercise 20 in chapter 2 - but this time i would need to Repeat Exercise 20 in Chapter 2 , using the ADT list to implement the function f (n). Consider the following recurrence relation: F (1) = 1; f (2) = 1; f(3) =1; f(4) =3; f(5) = 5; f (n) = f( n -1 ) + 3 x f(n-5) for all n > 5 Compute f(n ) for the following values of n : 6, 7, 12, 15. If you were careful, rather than computing f (15) from scratch (the way a recursive C++ function would compute it), you would have computed f (6), then f (7), then f(8), and so on up to f(15), recording the values as you computed them. This ordering would have saved you the effort of ever computing the same value more than once. (Recall the iterative version of the rabbit function discussed at the end of this chapter.) Note that during the computation
Using the ADT list, we can implement the function f(n) to compute the values of the recurrence relation F(n) = F(n-1) + 3 x F(n-5), for n > 5, where F(1) = 1, F(2) = 1, F(3) = 1, F(4) = 3, and F(5) = 5.
By computing the values of f(n) in an ordered manner, we can avoid computing the same value more than once. We can use this approach to compute f(6), f(7), f(12), and f(15).
We will use the ADT list to implement the function f(n) to compute the values of the recurrence relation F(n) = F(n-1) + 3 x F(n-5), for n > 5, where F(1) = 1, F(2) = 1, F(3) = 1, F(4) = 3, and F(5) = 5. We will compute the values of f(n) in an ordered manner to avoid computing the same value more than once.
First, we initialize an empty list to store the values of f(n). Then, we add the initial values of f(1), f(2), f(3), f(4), and f(5) to the list.
Next, we use a for loop to compute the values of f(n) for n = 6 to 15. Inside the for loop, we use the recurrence relation F(n) = F(n-1) + 3 x F(n-5) to compute the value of f(n). We check if the value of f(n) has already been computed by checking if the length of the list is greater than or equal to n.
If the value has already been computed, we skip the computation and move on to the next value of n. Otherwise, we add the computed value of f(n) to the end of the list.
After the for loop, the list contains the values of f(1) to f(15). We can extract the values of f(6), f(7), f(12), and f(15) from the list and print them out.
For example, the Python code to implement the above approach is:
# Initialize an empty list to store the values of f(n)
f_list = []
# Add the initial values of f(1), f(2), f(3), f(4), and f(5) to the list
f_list.extend([1, 1, 1, 3, 5])
# Compute the values of f(n) for n = 6 to 15
for n in range(6, 16):
if len(f_list) >= n:
# Value of f(n) has already been computed
continue
else:
# Compute the value of f(n) using the recurrence relation
f_n = f_list[n-2] + 3 * f_list[n-6]
# Add the computed value to the end of the list
f_list.append(f_n)
# Extract the values of f(6), f(7), f(12), and f(15) from the list
f_6 = f_list[5]
f_7 = f_list[6]
f_12 = f_list[11]
f_15 = f_list[14]
Print out the values of f(6), f(7), f(12), and f(15)
print("f(6)).
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Twenty-five students were asked to rate—on a scale of 0 to 10—how important it is to reduce pollution. A rating of 0 means “not at all important” and a rating of 10 means “very important.”
7 of the 25 measures are ratings of at most 6, that is, 6 is less than both the mean and the median of the distribution, hence it is not a good description of the center of this data set.
What is a data set?
A data set is a group of related data. In the case of tabular data, a data set relates to one or more database tables, where each row refers to a specific record in the corresponding data set and each column to a specific variable.
Here, we have
Given: Twenty-five students were asked to rate—on a scale of 0 to 10—how important it is to reduce pollution. A rating of 0 means “not at all important” and a rating of 10 means “very important.”
From the plot given in this exercise, it is found that only 7 of the 25 measures are ratings of at most 6, that is, 6 is less than both the mean and the median of the distribution.
Hence it is not a good description of the center of this data set.
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A park has grass and sand. Find the area of the part with grass.
(Sides meet at right angles.)
Answer:
[tex]26m^{2}[/tex]
Step-by-step explanation:
To solve this problem you find the total area of the entire rectangle and subtract the area of the sand from it. That will give you the area of the grass.
To find the total area you need to do [tex]b*h[/tex], in this case, the base is [tex]2+3+2[/tex] or 7. The height is 5. So to find the area, you have to multiply [tex]7*5[/tex] to get[tex]35m^{2}[/tex].
To find the area of the grass you multiply the [tex]b*h[/tex] or [tex]3*3[/tex] to get the area of 9.
Now the last step is to subtract [tex]35 - 9[/tex], doing so gives you your answer of 26 m
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find the exact value of cos 105 degrees by using the half-angle formula
The exact value of the cos 105 degrees by the using the half angle formula is -0.2588.
To find the exact value of cos(105°) using the half-angle formula, we first need to express 105° as half of another angle. Since 105° is equal to 210°/2,
we can use the half-angle formula for cosine:
cos(x/2) = ±√[(1 + cos(x))/2]
In our case, x = 210°. Now, we need to find the value of cos(210°):
210° lies in the third quadrant, where both sine and cosine are negative. To find the reference angle, subtract 180°:
210° - 180° = 30°
So, cos(210°) = -cos(30°) = -√3/2.
Now, let's plug the value of cos(210°) into the half-angle formula:
cos(105°) = ±√[(1 - √3/2)/2]
Since 105° lies in the second quadrant, where cosine is negative, we choose the negative root:
cos(105°) = -√[(1 - √3/2)/2]
=-0.2588
Explanation:- Here to find the value of the cosine 105 degree by using the half angle formula first we write the half angle formula of the cosine and substituted x=210 degree and simplify.
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Which equation has roots of +_ 3
From the list of options the equation with roots of ±3 is: (d) (x + 0)^2 = 3^2
Which equation has roots of +_ 3The equation that has roots of ±3 is:
(x - 3)(x + 3) = 0
Expanding the left side of the equation using FOIL method, we get:
x^2 - 9 = 0
Therefore, the equation with roots of ±3 is:
x^2 - 9 = 0
Add 9 to both sides
x^2 = 9
Express 9 as 3^2
x^2 = 3^2
So, we have
(x + 0)^2 = 3^2
Therefore, the equation with roots of ±3 is: (d) (x + 0)^2 = 3^2
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In Exercises 25-29, the given set is a subset of C[−1,1]. Which of these are also vector spaces? 1F= {f(x) in C[−1,1] : ∫ f(x)dx=0}−1
The given set F = {f(x) in C[-1,1] : ∫ f(x)dx = 0 from -1 to 1} is a vector space.
To verify if F is a vector space, we need to check if it satisfies the vector space axioms. Let f(x) and g(x) be elements of F, and c be a scalar.
1. Closure under addition: ∫ (f(x) + g(x))dx = ∫ f(x)dx + ∫ g(x)dx = 0 + 0 = 0. So, (f(x) + g(x)) is in F.
2. Closure under scalar multiplication: ∫ (cf(x))dx = c∫ f(x)dx = c(0) = 0. So, (cf(x)) is in F.
3. Contains zero vector: The zero function, f(x) = 0, satisfies ∫ f(x)dx = 0, and is in F.
Since F satisfies these axioms, it is a vector space.
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The given set F = {f(x) in C[-1,1] : ∫ f(x)dx = 0 from -1 to 1} is a vector space.
To verify if F is a vector space, we need to check if it satisfies the vector space axioms. Let f(x) and g(x) be elements of F, and c be a scalar.
1. Closure under addition: ∫ (f(x) + g(x))dx = ∫ f(x)dx + ∫ g(x)dx = 0 + 0 = 0. So, (f(x) + g(x)) is in F.
2. Closure under scalar multiplication: ∫ (cf(x))dx = c∫ f(x)dx = c(0) = 0. So, (cf(x)) is in F.
3. Contains zero vector: The zero function, f(x) = 0, satisfies ∫ f(x)dx = 0, and is in F.
Since F satisfies these axioms, it is a vector space.
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Question 18 of 26
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Given
�
(
�
)
=
−
2
�
−
2
f(x)=−2x−2, find
�
−
1
(
�
)
f
−1
(x).
The value of f ⁻¹ (x) would be,
⇒ f⁻¹(x) = - x/2 - 1
We have to given that;
Function is,
⇒ f (x) = - 2x - 2
Now, We can find the inverse of function as;
⇒ f (x) = - 2x - 2
⇒ y = - 2x - 2
Solve for x;
⇒ y + 2 = - 2x
⇒ x = - (y + 2)/2
⇒ x = - y/2 - 1
⇒ f⁻¹(x) = - x/2 - 1
Thus, The value of f ⁻¹ (x) would be,
⇒ f⁻¹(x) = - x/2 - 1
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What will be the graph of the function f(x) = 2x + 26
The graph of the function f(x) = 2x + 26 will be a line graph.
Some key points about the graph:
• The slope of the line will be 2.
• The y-intercept will be 26, since f(0) = 26.
• The line will pass through the points (0, 26) and (x, 2x + 26).
• As x increases, the value of f(x) also increases but at a increasing rate.
• The graph will be a positively sloped line, increasing from left to right.
A rough sketch of the graph would be:
y
26
24
22
20
18
16
14
12
10
8
6
4
2
-6 -4 -2 0 2 4 6 8 10 12 14 x
Does this help explain the graph? Let me know if you have any other questions!
The power P
in a motor is given by the formula P=IV
I=
current, V=
voltage. Find P
when I=64. 1
, V=12. 8
The power (P) when I = 64.1 A and V = 12.8 V is 819.68 watts (W).
Hello! I understand that you need help in finding the power (P) when the current (I) is 64.1 A and the voltage (V) is 12.8 V.
Understand the relationship between power, current, and voltage.
The power (P) in an electrical circuit can be calculated using the formula P = I × V,
where I is the current in amperes (A) and V is the voltage in volts (V).
Plug in the given values.
In this case, we are given I = 64.1 A
and V = 12.8 V.
Plug these values into the formula:
P = 64.1 A × 12.8 V
Calculate the power.
Multiply the current and voltage to find the power:
P = 819.68 W.
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What is the domain of this function?
(-3, -7)
(-8, 8)
(10, 1)
(9,4)
(7,-4)
(-6, -9)
Answer: (-8, -6, -3, 7, 9, 10)
Step-by-step explanation:
The domain of a function is the set of all possible x-values that correspond to the given ordered pairs.
Therefore, the domain of the function is: (-8, -6, -3, 7, 9, 10)
Solve the separable differential equation for u, du/dt = e^5u+5t. Use the following initial condition: u(0) = 15.
Th solution of the given seperable differential equation is :
u = (1/5)ln(25t^2 + e^75)
To solve the separable differential equation for u, we need to separate the variables and integrate both sides.
First, we can write the equation as:
(1/e^5u)du = 5t dt
Now we can integrate both sides:
∫(1/e^5u)du = ∫5t dt
To integrate the left side, we can use u-substitution:
Let u = 5u
Then du = 5e^5u du
Substituting into the integral, we get:
(1/5)∫e^5u du = ∫5t dt
(1/5)e^5u = 5t^2/2 + C
Where C is the constant of integration.
Now we can solve for u:
e^5u = 25t^2 + 2C
Taking the natural logarithm of both sides:
5u = ln(25t^2 + 2C)
u = (1/5)ln(25t^2 + 2C)
Using the initial condition u(0) = 15, we can solve for C:
15 = (1/5)ln(2C)
ln(2C) = 75
2C = e^75
C = (1/2)e^75
Substituting this value of C into our solution for u, we get:
u = (1/5)ln(25t^2 + e^75)
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suppose that n(u ) = 200 , n(e ∪ f ) = 194 , n(e) = 106 , and c n(e ∩ f ) = 73 . find each of the following values. n (e ∪ f)c
N(e ∪ f) = 194.
Using the inclusion-exclusion principle, we have:
n(e ∪ f) = n(e) + n(f) - n(e ∩ f)
We are given n(e ∩ f) = 73 and n(e ∪ f) = 194, so we can rearrange to solve for n(f):
n(f) = n(e ∪ f) - n(e) + n(e ∩ f)
n(f) = 194 - 106 + 73
n(f) = 161
Finally, to find n(e ∪ f), we can substitute the values we have found into the first equation:
n(e ∪ f) = n(e) + n(f) - n(e ∩ f)
n(e ∪ f) = 106 + 161 - 73
n(e ∪ f) = 194
Therefore, n(e ∪ f) = 194.
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Suppose a certain country's population has constant relative birth and death rates of 97 births per thousand people per year and 47 deaths per thousand people per year respectively. Assume also that approximately 30,000 people emigrate (leave) from the country every year. Which equation best models the population P- Pt) of the country, where t is in years? dP - 50P30000 Oat 30000 dit d0.05P30000 it 0.5P30000 dP 0.5P 30 ait
The equation that best models the population P(t) of the country, given the constant relative birth and death rates and the number of people emigrating every year, is dP/dt = 0.5P - 30,000.
The rate of population growth is determined by the difference between the birth rate and the death rate, which is (97 - 47) per thousand people per year, or 0.05. This means that the population will grow by 0.05 times the current population each year if there is no emigration. However, since 30,000 people emigrate every year, we need to subtract this number from the population growth rate. Therefore, the rate of population growth can be expressed as 0.05P - 30,000.
To get the population at any given time t, we need to integrate this rate equation concerning time t. The solution to this differential equation is P(t) = (P0 - 60,000)e^(0.05t) + 60,000, where P0 is the initial population at time t=0.
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Jetia mixes 5 parts cranberry juice with 8 parts apple juice to make 177 cups of
Answer: 108 cups of cranberry juice. Brainliest?
Step-by-step explanation:
mixed juice. How many cups of cranberry juice did Jetia use?
Let's start by assuming that Jetia used x cups of cranberry juice to make the mixed juice. Then, since the ratio of cranberry juice to apple juice is 5:8, she must have used (5/8)x cups of apple juice.
We know that the total amount of mixed juice is 177 cups, so we can set up an equation based on the total amount of juice:
x + (5/8)x = 177
Simplifying this equation, we get:
(13/8)x = 177
Multiplying both sides by 8/13, we get:
x = 108
Therefore, Jetia used 108 cups of cranberry juice to make the mixed juice.