Verify the following Pythagorean identity for all values of x and y:
(x² + y²)² = (x² - y²)² + (2xy)²

This means that the coordinates of P are (a, b), where a is the value for which f'(a) = 0, and b = f(a).

The term "coordinate" generally refers to a value or set of values that describe the position or location of a point or object in a particular system or space. In mathematics, coordinates are typically used to describe the position of a point on a graph or in a geometric plane, and they usually consist of a set of numerical values that indicate the distance or direction of the point from a specified origin or reference point. In geographic or cartographic contexts, coordinates might refer to latitude and longitude values that indicate a specific location on the Earth's surface. Other systems may use different types of coordinates, but the underlying idea is usually the same: a set of values that describe the position of an object or point in a given space or system.

if the graph of f has a horizontal tangent at point P, this means that the slope of the tangent line at P is zero.

Let (a, b) be the coordinates of P. Then the equation of the tangent line at P is:

[tex]y - b = f'(a)(x - a)[/tex]

Since the slope of the tangent line at P is zero, we have:

f'(a) = 0

This means that the derivative of f at x = a is zero. So, we need to find the value of a for which f'(a) = 0.

Once we find the value of a, we can substitute it into the equation of the tangent line to find the value of b.

So, let's find f'(x) first:

f(x) = ... (the function is not given, so we cannot find f'(x) explicitly)

We know that f'(a) = 0, so we have:

[tex]f'(a) = lim h- > 0 [f(a+h) - f(a)]/h = 0[/tex]

This means that:

[tex]lim h- > 0 [f(a+h) - f(a)]/h = 0[/tex]

Multiplying both sides by h, we get:

[tex]lim h- > 0 [f(a+h) - f(a)] = 0[/tex]

Taking the limit as h approaches 0, we get:

[tex]f(a) - f(a) = 0[/tex]

So, we have:

[tex]f(a) = b[/tex]

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The graph that represents the equation is:

The graph shows an upward parabola with vertex (2, -6) and passes through (-1, 7), (0, 0), (4, 0), and (5, 7).

Option B is the correct answer.

We have,

The equation y = (3/2)x² - 6x represents an upward parabola since the coefficient of x² is positive.

Now,

The vertex of the parabola.

x = -b/2a,

where a and b are the coefficients of x² and x, respectively.

So,

a = 3/2 and b = -6

x = -(-6)/(2(3/2)) = 6/3 = 2.

Plugging x = 2 into the equation,

We get y = 3/2(2)² - 6(2) = -6,

so the vertex is (2, -6).

We can eliminate options A and C since their vertices are not (2, -6).

Now,

To check which of the remaining options fits the equation, we can plug in the given points and see if they satisfy the equation.

Option B gives:

When x = -1, y = 3/2(-1)² - 6(-1) = 7

When x = 0, y = 3/2(0)² - 6(0) = 0

When x = 4, y = 3/2(4)² - 6(4) = 0

When x = 5, y = 3/2(5)² - 6(5) = 7.5

So option B fits the equation, and is the graph that represents

y = (3/2)x² - 6x.

Therefore,

The graph that represents the equation is:

The graph shows an upward parabola with vertex (2, -6) and passes through (-1, 7), (0, 0), (4, 0), and (5, 7).

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For an isosceles triangle, a leg is congruent to the other leg. A leg is not necessarily congruent to the base. The correct solution found by solving the equation 2x = 3x - 5 is x = 5. The length of the top left side is 10, and the length of the bottom left side is 10.

In Mathematics and Geometry, an isosceles triangle simply refers to a type of triangle with two (2) sides that are equal in length and two (2) equal angles.

This ultimately implies that, an isosceles triangle comprises two (2) side lengths that are equal and two (2) angles with the same magnitude.

In this isosceles triangle, we have the following equation;

2x = 3x - 5 (two (2) side lengths are congruent).

3x - 2x = 5

x = 5 units.

For the bottom left side, we have;

3x - 5 = 3(5) - 5 = 15 - 5 = 10 units.

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Answer:

y = [tex]\sqrt{11}[/tex]

Step-by-step explanation:

Substitute -4 for x and solve for y

[tex]y^{2}[/tex] = -2x + 3

[tex]y^{2}[/tex] = -2(-4) + 3

[tex]y^{2}[/tex] = 8 + 3

[tex]y^{2}[/tex] = 11

[tex]\sqrt{y^{2} }[/tex] = [tex]\sqrt{11}[/tex]

y = [tex]\sqrt{11}[/tex]

Helping in the name of Jesus.

Therefore, after 8 years, the interest will be $20,146.60.

When a customer pays interest to borrow money from a bank, for example, interest is referred to as a payment from the borrower in the finance industry.

The customer would pay an amount that is greater than the amount they borrowed due to interest.

The principal is the sum of money that was first invested or lent and upon which interest is based.

We refer to the process of calculating fresh interest based on the new principal (which is the old principal plus interest) at the conclusion of the subsequent payment period as compound interest when we add interest to interest.

So, compound interest is interest that is paid on both the principal and interest that has already been generated.

We may calculate compound interest using the following straightforward formula:

That has to be clarified in order for us to address the issue at hand:

17400 invested at a 2.5% annual compounded rate for an 8-year period.

Let's take a look at what is provided and note it down.

Therefore, the principal is $43,000.00. In addition, we know that our rate is 2.5%, or 0.025

If it is annually compounded, n=1 times every year.

Additionally, t=8 years.

A = 17400(1 + 0.025/1)⁸ A = $20,146.60

Therefore, after 8 years, the balance will be $20,146.60.

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Therefore, after 8 years, the interest will be $20,146.60.

When a customer pays interest to borrow money from a bank, for example, interest is referred to as a payment from the borrower in the finance industry.

The customer would pay an amount that is greater than the amount they borrowed due to interest.

The principal is the sum of money that was first invested or lent and upon which interest is based.

We refer to the process of calculating fresh interest based on the new principal (which is the old principal plus interest) at the conclusion of the subsequent payment period as compound interest when we add interest to interest.

So, compound interest is interest that is paid on both the principal and interest that has already been generated.

We may calculate compound interest using the following straightforward formula:

That has to be clarified in order for us to address the issue at hand:

17400 invested at a 2.5% annual compounded rate for an 8-year period.

Let's take a look at what is provided and note it down.

Therefore, the principal is $43,000.00. In addition, we know that our rate is 2.5%, or 0.025

If it is annually compounded, n=1 times every year.

Additionally, t=8 years.

A = 17400(1 + 0.025/1)⁸ A = $20,146.60

Therefore, after 8 years, the balance will be $20,146.60.

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The most common amount of sleep is 8 1/2 hours.

There is no outlier in the data.

The difference in hours between the longest night's sleep and the longest night's sleep is 3.5 hours.

The number of people who slept longer than 8 hours is 15 people.

In Mathematics and Statistics, a dot plot can be defined as a type of line plot that is typically used for the graphical representation of a data set above a number line, especially through the use of crosses or dots.

Based on the information provided about the number of hours Molly's friends slept last night, we can reasonably infer and logically deduce that all of the data points would be between 6 and 11, without any outlier.

In this scenario, we would use an online graphing calculator to construct a dot plot with respect to a number line that accurately fit Molly's data set, with a scale of 0 < x < 11.

Difference in hours = 10 - 6.5 = 3.5 hours.

In conclusion, the dot plot would be titled "Hours of sleep."

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Missing information:

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

To calculate the gross profit, we need to subtract the cost of goods sold (COGS) from the revenue generated by sales. The revenue generated by sales can be found by multiplying the number of printers sold by the selling price of each printer, which is $1,750.

Revenue generated by sales = 26 x $1,750 = $45,500

The cost of goods sold for each printer is $975, so the total cost of goods sold for all 26 printers is:

Total cost of goods sold = 26 x $975 = $25,350

Therefore, the gross profit can be calculated as:

Gross profit = Revenue generated by sales - Total cost of goods sold

Gross profit = $45,500 - $25,350

Gross profit = $20,150

The gross profit is $20,150.

To calculate the net income, we need to subtract all operating expenses from the gross profit. The operating expenses include your salary ($3,000/month), advertising ($50/month), and office lease ($550/month).

Total operating expenses = $3,000 + $50 + $550

Total operating expenses = $3,600

Therefore, the net income can be calculated as:

Net income = Gross profit - Total operating expenses

Net income = $20,150 - $3,600

Net income = $16,550

The net income is $16,550.

In summary, the gross profit is the revenue generated by sales minus the cost of goods sold, which in this case is $20,150. The net income is the gross profit minus all operating expenses, which in this case is $16,550.

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The solution to the equations for y are y = 9, y = 8 and y = 7

From the question, we have the following parameters that can be used in our computation:

y²- 1 = 80

(y - 4)÷ 4 = 1

10 - 63 ÷ y = 1​

Solving the equations, we have

y²- 1 = 80

y² = 81

y = 9

Next, we have

(y - 4)÷ 4 = 1

(y - 4) = 4

y = 8

Lastly, we have

10 - 63 ÷ y = 1​

-63 ÷ y = -9

-9y = -63

y = 7

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Okay, let's analyze the linearized data options to determine which equation best models it:

a. log y = 0.116315x+0.249114

This has a positive slope coefficient of 0.116315, so it will model increasing data. But the y-intercept term 0.249114 is negative, so the model line would start below the origin. This does not seem to match the typical linear relationship we would expect.

b. log y = 0.249114x+0.116315

This also has a positive slope coefficient, but now the y-intercept term 0.116315 is positive. This could potentially model increasing data that starts above the origin. However, without seeing the data it's hard to definitively say if this is the best fit.

c. log y = 0.0288761x+0.435269

This has a positive but much smaller slope coefficient of 0.0288761. This would model increasing data at a much shallower rate. Again, without seeing the data we can't rule this out, but it seems less likely.

d. log y = 0.435269x+0.0288761

This has a larger positive slope coefficient of 0.435269, so it would model increasing data at a steeper rate. And the positive y-intercept term of 0.0288761 ensures the model line starts above the origin.

Of these 4 options, choice d (log y = 0.435269x+0.0288761) seems the most likely to best model increasing linear data that starts above the origin. However, without seeing the actual data points, we can't say that with certainty. The slope and intercept values would need to give the closest fit to the data for that to be the definitive choice.

Let me know if this helps explain the options, or if you have any other questions!

Answer:

Yes, you did solve it correctly, and the way you did it is fine

Step-by-step explanation:

You correctly applied the formula for the Law of Cosines to solve for the unknown side length c of a triangle, given the values of the other two side lengths and the angle opposite the unknown side. Specifically, you plugged in the given values of a, b, and C into the formula c^2 = a^2 + b^2 - 2ab cos(C) and then solved for c by taking the square root of both sides of the equation. Finally, you simplified the expression for c by calculating the square root of 25 to obtain a numerical value of c=5

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The shortest length that rope can be, is 360 cm; such that the rope may be cut into lengths of exactly 18cm long, 30cm long or 40cm long.

A rope is cut into length which may be exactly 18cm long, 30cm long or 40cm long.

The shortest length that rope can be

Considering the individual lengths which are all integers, the shortest length is calculated as the LCM or the Least Common Multiple of the individual lengths that are considered

Factorize the numbers 18, 30 and 40 in the following way.

18 = 2 * 3 * 3

30 = 2* 3 *5

40 = 2* 2* 2* 5

The LCM of  the numbers 18, 30 and 40 is

= 2* 2* 2* 3* 3* 5

= 360

So, the shortest length that rope can be, to make it cut into lengths of exactly 18cm long, 30cm long or 40cm long, is 360 cm.

Therefore, the required shortest length that rope can be, is 360 cm; such that the rope may be cut into lengths of exactly 18cm long, 30cm long or 40cm long.

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Front-end rounding 10,355 to the nearest ten is 10,360.

Rounding is the process of approximating a number to a specified degree of accuracy or precision. Rounding is useful when we want to simplify numbers or express them in a more manageable or meaningful way. There are various methods of rounding, such as rounding up, rounding down, and rounding to the nearest value.

To round a number to a specified degree of accuracy, we first identify the place value we are rounding to, such as units, tens, hundreds, etc. We then look at the digit in that place value and the digit to the right of it. If the digit to the right is 5 or greater, we round up by adding 1 to the digit in the place value we are rounding, and then replace all the digits to the right with zeros. If the digit to the right is less than 5, we round down by simply dropping all the digits to the right.

To front-end round a number, you look at the digit in the place value you are rounding to and round up or down based on the digit to the right of that place value.

To front-end round 10,355 to the nearest ten, we look at the tens place, which is the second digit from the right. The digit in the tens place is 5, which is greater than or equal to 5, so we round up.

To round up, we add 1 to the digit in the tens place and replace all the digits to the right with zeros. So, rounding up 10,355 to the nearest ten gives us:

10,360

Therefore, front-end rounding 10,355 to the nearest ten is 10,360.

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Answer:

Step-by-step explanation:

You want to determine the number of solutions to three different systems of equations.

A linear equation is written in standard form when the coefficients are mutually prime, and the leading coefficient is positive:

  ax +by = c . . . . . . GCF(a, b, c) = 1, a > 0

The equations are easiest to compare when they are all written in standard form.

A system will have an infinite number of solutions when the equations are identical.

A system will have zero solutions when it reduces to ...

  (non-zero constant) = 0

A system will have one solution when the equations are different.

A factor of 2 can be removed from the first equation:

  2x -3y = 5

A factor of 3 can be removed from the second equation:

  2x -3y = 5

These equations are identical, so have infinitely many solutions.

Multiplying the first equation by 2 gives ...

  2y = -3x +6

Adding 3x, we have ...

  3x +2y = 6

When we subtract the second equation from this, we get ...

  (3x +2y) -(3x +2y) = (6) -(3)

  0 = 3

These equations have no solution.

These equations are already in standard form, and are different. This system has one solution.

(The exact solution is (x, y) = (0.48, 3.36).)

Answer:

60.5 m

Step-by-step explanation:

We can use the arc length formula to calculate the length of highlighted arc. Let the numerical measure of the highlighted arc be known as C, the angle measure be known as [tex]\theta[/tex], and r as the radius of the circle.

Given variables:

Plug in all the variables into the formula and solve for the letter C.

[tex]\boxed{\text{Formula: C = }{2\pi r\huge{\text(}\frac{\theta}{360}\huge\text{)}}}[/tex]

[tex]\implies C = }{2\pi (11)\huge{\text(}\dfrac{315}{360}\huge\text{)}}}[/tex]   [tex]\implies C = }{22\pi \huge{\text(}\dfrac{315}{360}\huge\text{)}}} = 60.5 \ \text{m}[/tex]

Therefore, Option A is the correct option.

The probabilities are given as follows:

4. A. 1/2.

5. B. 3/7.

6. C. 3/13.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

For item 4, we have that out of 30 trials, an Alaskan Malamute won 15 times, hence the probability is given as follows:

p = 15/30

p = 1/2.

For item 5, we have that out of 42 trials, a Siberian Husky was chosen 18 times, hence the probability is given as follows:

p = 18/42

p = 3/7.

For item 6, we have that out of 52 cards in a deck, 12 are pictures, hence the probability is given as follows:

p = 12/52

p = 3/13.

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Answer:

1 = y^2 - 2

y^2 = 3, so y = -√3

The maximum price you should be willing to pay for the annuity, given the cash flow and the interest rates, would be 2. R 39, 272

The maximum price to be paid for the annuity would be the present value of the annuity which can be found by the formula :

= Annuity x Present value interest factor of annuity, 20 years, 8 %

The annuity = R 4, 000

Present value interest factor of annuity, 20 years, 8 %  = 9. 81815

The maximum you should pay for the annuity is therefore:

= 4, 000 x 9. 81815

=  R 39, 272 . 60

= R 39, 272

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Answer:

9x + 13x = 176

22x = 176

x = 8

So the numbers are 9 × 8 = 72 and

13 × 8 = 104.

The function that would match the graph shown below is given as follows:

y = -5(x³ + 7x² + 8x - 16).

The function is defined using the Factor Theorem, as we have the x-intercepts of the graph, hence we can write the function as a product of it's linear factors.

Considering the x-intercepts, the roots are given as follows:

Hence, considering the leading coefficient a, the function is defined as follows:

y = a(x + 4)²(x - 1)

y = a(x² + 8x + 16)(x - 1)

y = a(x³ + 7x² + 8x - 16).

When x = 0, y = 80, hence the leading coefficient a is obtained as follows:

-16a = 80

a = -5.

Hence the function is:

y = -5(x³ + 7x² + 8x - 16).

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Answer:

4/5

Step-by-step explanation:

2 : 5/2

4/2 : 5/2

4 : 5

The scale factor is:

4/5

Any point that includes the origin that comes in the shaded region will satisfy the system of equation.

For the points that satisfy the system of inequalities y < x² + 3 and y > x² - 4, we need to graph the two parabolas y = x² + 3 and y = x² - 4 and shade the region where the two graphs overlap.

The graph of y = x² + 3 is an upward-facing parabola that intersects the y-axis at (0, 3). The graph of y = x² - 4 is also an upward-facing parabola that intersects the y-axis at (0, -4).

We can see from the graphs that the region where y < x² + 3 and y > x² - 4 is the shaded region between the two parabolas. This region is bounded by the two lines y = x² + 3 and y = x² - 4.

To find a point that satisfies this system of inequalities, we can pick any point within this shaded region. For example, the point (0, 0) lies within the shaded region and satisfies both inequalities:

[tex]y = 0 < x^2+ 3 = 3\\y = 0 > x^2 - 4 = -4[/tex]

Therefore, the point (0, 0) is a solution to the system of inequalities.

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The zeros of the function f(x) = (-x - 2)(-2x - 3) are x = -2 and x = -3/2, listed from least to greatest.

Given the function in the question:

f(x) = ( -x - 2 )( -2x - 3 )

To determine the zeros of the function f(x), we need to solve the equation f(x) = 0.

f(x) = ( -x - 2 )( -2x - 3 )

0 = ( -x - 2 )( -2x - 3 )

( -x - 2 )( -2x - 3 ) = 0

We can set each factor equal to zero and solve for x:

( -x - 2 ) = 0

-x - 2 = 0

-x = 2

x = -2

( -2x - 3 ) = 0

-2x - 3 = 0

-2x = 3

x = -3/2

Therefore, the zeros are x = -2 and x = -3/2.

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The equation of line L is given as follows:

L: x = 2.

The equation of line L is the line of symmetry of the quadratic function given as follows:

y = 3x² - 12x + 7.

Considering a quadratic function with equation y = ax² + bc + c, the line of symmetry is defined as follows:

L: x = -b/2a.

The coefficients of the function are given as follows:

a = 3, b = -12.

Hence the equation of line L is defined as follows:

L: x = -(-12)/2(3)

L: x = 2.

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The logarithm of 6.373log4.948/sqrt(0.004636) is approximately 2.2022

Logarithm is the inverse operation of exponentiation. It is a function that tells you what power you need to raise a given base to in order to get a certain value.

The logarithm of a number x with respect to  base a is denoted as logₐ(x).

For example, if we have a base of 2 and a value of 8, we can write:

log₂(8) = 3

Going back to our question, we can apply the basic knowledge of logarithm to solve the question.

First, simplify the expression:

6.373log4.948/sqrt(0.004636)

= 6.373 * log(4.948) / sqrt(0.004636)

= 6.373 * 1.6941 / 0.068

= 159.50

Now, we can find the logarithm of 159.50.

log(159.50) ≈ 2.2022

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An equation to match the graph include the following: f(x) = |x - 1| - 1.

In Mathematics and Geometry, the translation a geometric figure or graph to the right simply means adding a digit to the value on the x-coordinate of the pre-image while the translation a geometric figure or graph downward simply means subtracting a digit from the value on the y-coordinate (y-axis) of the pre-image.

Since the parent function is f(x) = |x|, g(x) would be created by translating f(x) the parent function one units to the right and one units downward as follows;

f(x) = |x|

g(x) = |x - 1| - 1

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Answer:

the answer is E

Step-by-step explanation: