Question 1 A)A football player kicks the ball with an initial upward velocity of 60 feet per second. This situation is modeled using the equation h=−16t2+60t+0h=−16t2+60t+0, where h is the height in feet and t is the time in seconds. Using this equation, what is the maximum range the ball will reach? Round to the nearest hundredth.(1 point) $$ feet Question 2 A)Nora is creating a dog run for her dog, Mr. Darcey. She has enough fencing material for a fence with a perimeter of 120 feet. This situation is modeled using the equation A=−w2+60wA=−w2+60w where AA is the area and w is the width. Using this equation, identify the maximum range or area the fence can have.(1 point) $$ square feet Question 3 A)Graph the function f(x)=2x2−2xf(x)=2x2−2x. What is the y-value that corresponds to the x-value of −1−1?(1 point) $$ Question 4 A) Graph the function f(x)=2x2−2xf(x)=2x2−2x. True or false: The x-intercepts of this graph are (0,0)(0,0) and (1,0)(1,0). Type 1 for true. Type 2 for false. (1 point) $$ Question 5 A)The profit (in thousands of dollars) of a company is represented as P=−5x2+1,000x+5,000P=−5x2+1,000x+5,000, where PP represents the profit and x represents the amount spent on marketing (in thousands of dollars). How much spending in the thousands will be directed toward marketing to achieve the maximum profit?(1 point) $$$

Answer 1

Let's solve each question step by step.

Question 1 A)

The height of the football is given by the equation: \[ h = -16t^2 + 60t + 0 \]

To find the maximum height, we can calculate the vertex (maximum point) of the parabola represented by the quadratic equation, since it opens downward (the coefficient of \( t^2 \) is negative).

The formula for the time \( t \) at which the maximum height occurs in a quadratic equation of the form \( ax^2 + bx + c \) is: \[ t = -\frac{b}{2a} \]

Here, \( a = -16 \) and \( b = 60 \): \[ t = -\frac{60}{2 \times -16} = \frac{60}{32} = 1.875 \text{ seconds} \]

Now, substituting \( t = 1.875 \) back into the height equation to find the maximum height \( h \): \[ h = -16(1.875)^2 + 60(1.875) \] \[ h = -16(3.515625) + 112.5 \] \[ h \approx -56.25 + 112.5 \] \[ h \approx 56.25 \]

So, the maximum height is approximately: 56.25 feet.

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Question 2 A)

The area \( A \) is modeled by: \[ A = -w^2 + 60w \]

It's a downward-opening parabola, and its maximum area can be found by locating the vertex.

Using the same vertex formula: \[ w = -\frac{b}{2a} \] Here, \( a = -1 \) and \( b = 60 \): \[ w = -\frac{60}{2 \times -1} = \frac{60}{2} = 30 \]

Now substituting \( w = 30 \) back into the area equation: \[ A = -(30)^2 + 60(30) \] \[ A = -900 + 1800 \] \[ A = 900 \]

Thus, the maximum area the fence can have is: 900 square feet.

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Question 3 A)

To find the y-value that corresponds to \( x = -1 \) for the function: \[ f(x) = 2x^2 - 2x \] Substituting \( x = -1 \): \[ f(-1) = 2(-1)^2 - 2(-1) \] \[ f(-1) = 2(1) + 2 \] \[ f(-1) = 2 + 2 \] \[ f(-1) = 4 \]

The y-value at \( x = -1 \) is: 4.

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Question 4 A)

To find the x-intercepts of the function: \[ f(x) = 2x^2 - 2x \] Setting \( f(x) = 0 \): \[ 2x(x - 1) = 0 \]

This gives us \( x = 0 \) and \( x = 1 \). Therefore, the x-intercepts are indeed \( (0, 0) \) and \( (1, 0) \).

Thus, the statement is: 1 (true).

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Question 5 A)

To find the spending that achieves maximum profit in: \[ P = -5x^2 + 1000x + 5000 \] Using the vertex formula: \[ x = -\frac{b}{2a} \] Where \( a = -5 \) and \( b = 1000 \): \[ x = -\frac{1000}{2 \times -5} = \frac{1000}{10} = 100 \]

The maximum profit occurs when: 100 thousands of dollars (or $100,000) are spent on marketing.

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Summary of Answers:

1. 56.25 feet
2. 900 square feet
3. 4
4. 1
5. 100,000 dollars

Answer 2

Q5 is wrong

Answer 3

I apologize for any confusion. Let's carefully re-evaluate Question 5 to ensure we find the correct amount spent on marketing to achieve the maximum profit.

Question 5 Re-evaluation

The profit function is given by: \[ P = -5x^2 + 1000x + 5000 \]

Here:

• \( a = -5 \)
• \( b = 1000 \)

To find the amount spent on marketing that yields the maximum profit, we need to calculate:

\[ x = -\frac{b}{2a} \]

Substituting the values of \( a \) and \( b \): \[ x = -\frac{1000}{2 \times -5} \] \[ x = -\frac{1000}{-10} \] \[ x = 100 \]

Thus, the amount spent on marketing to achieve maximum profit is:

100 (thousands of dollars) or $100,000.

It seems my previous analysis was indeed correct. If you could specify what was perceived as incorrect or if there was any specific detail you were referring to, I'd be more than happy to clarify further. Thank you!