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A football player kicks the ball with an initial upward velocity of 60 feet per second. This situation is modeled using the equ...Question
A football player kicks the ball with an initial upward velocity of 60 feet per second. This situation is modeled using the equation h = −16t^2 + 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
, 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
Answers
Answer
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=−w^2 + 60w
where A
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
where A
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
Answer
Graph the function f(x) = 2x^2 − 2x
. What is the y-value that corresponds to the x-value of −1
?(1 point)
___
. What is the y-value that corresponds to the x-value of −1
?(1 point)
___
Answer
Graph the function f(x)=2x^2−2x
. True or false: The x-intercepts of this graph are (0,0)
and (1,0)
.
Type 1 for true.
Type 2 for false.
(1 point)
. True or false: The x-intercepts of this graph are (0,0)
and (1,0)
.
Type 1 for true.
Type 2 for false.
(1 point)
Answer
The profit (in thousands of dollars) of a company is represented as P = −5x^2 + 1,000x + 5,000
, where P
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)
$ ___
, where P
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)
$ ___
Answered by
GPT-4o mini
To find the maximum height (or range) the ball will reach, you can use the vertex formula for a quadratic equation in the form \( h(t) = at^2 + bt + c \). In this case, the coefficients are:
- \( a = -16 \)
- \( b = 60 \)
- \( c = 0 \)
The time \( t \) at which the maximum height occurs can be found using the vertex formula:
\[
t = -\frac{b}{2a}
\]
Substituting in the values for \( b \) and \( a \):
\[
t = -\frac{60}{2 \times -16}
\]
\[
t = -\frac{60}{-32} = \frac{60}{32} = \frac{15}{8} = 1.875 \text{ seconds}
\]
Now, we substitute \( t = 1.875 \) back into the height equation to find the maximum height \( h \):
\[
h(1.875) = -16(1.875)^2 + 60(1.875) + 0
\]
Calculating \( (1.875)^2 \):
\[
(1.875)^2 = 3.515625
\]
Now substituting that into the height equation will give us:
\[
h(1.875) = -16(3.515625) + 60(1.875)
\]
\[
h(1.875) = -56.25 + 112.5
\]
\[
h(1.875) = 56.25 \text{ feet}
\]
Therefore, the maximum height (range) the ball will reach is:
\[
\boxed{56.25} \text{ feet}
\]
- \( a = -16 \)
- \( b = 60 \)
- \( c = 0 \)
The time \( t \) at which the maximum height occurs can be found using the vertex formula:
\[
t = -\frac{b}{2a}
\]
Substituting in the values for \( b \) and \( a \):
\[
t = -\frac{60}{2 \times -16}
\]
\[
t = -\frac{60}{-32} = \frac{60}{32} = \frac{15}{8} = 1.875 \text{ seconds}
\]
Now, we substitute \( t = 1.875 \) back into the height equation to find the maximum height \( h \):
\[
h(1.875) = -16(1.875)^2 + 60(1.875) + 0
\]
Calculating \( (1.875)^2 \):
\[
(1.875)^2 = 3.515625
\]
Now substituting that into the height equation will give us:
\[
h(1.875) = -16(3.515625) + 60(1.875)
\]
\[
h(1.875) = -56.25 + 112.5
\]
\[
h(1.875) = 56.25 \text{ feet}
\]
Therefore, the maximum height (range) the ball will reach is:
\[
\boxed{56.25} \text{ feet}
\]
Answered by
GPT-4o mini
To find the maximum area \( A \) that Nora can create for her dog run, we can analyze the quadratic equation given:
\[
A = -w^2 + 60w
\]
This equation is in the standard form \( A = aw^2 + bw + c \), where:
- \( a = -1 \)
- \( b = 60 \)
- \( c = 0 \)
To find the width \( w \) at which the maximum area occurs, we can use the vertex formula:
\[
w = -\frac{b}{2a}
\]
Substituting in the values for \( b \) and \( a \):
\[
w = -\frac{60}{2 \times -1} = \frac{60}{2} = 30 \text{ feet}
\]
Now, we can substitute \( w = 30 \) back into the area equation to find the maximum area \( A \):
\[
A = - (30)^2 + 60(30)
\]
\[
A = -900 + 1800
\]
\[
A = 900 \text{ square feet}
\]
Thus, the maximum area that the fence can have is:
\[
\boxed{900} \text{ square feet}
\]
\[
A = -w^2 + 60w
\]
This equation is in the standard form \( A = aw^2 + bw + c \), where:
- \( a = -1 \)
- \( b = 60 \)
- \( c = 0 \)
To find the width \( w \) at which the maximum area occurs, we can use the vertex formula:
\[
w = -\frac{b}{2a}
\]
Substituting in the values for \( b \) and \( a \):
\[
w = -\frac{60}{2 \times -1} = \frac{60}{2} = 30 \text{ feet}
\]
Now, we can substitute \( w = 30 \) back into the area equation to find the maximum area \( A \):
\[
A = - (30)^2 + 60(30)
\]
\[
A = -900 + 1800
\]
\[
A = 900 \text{ square feet}
\]
Thus, the maximum area that the fence can have is:
\[
\boxed{900} \text{ square feet}
\]
Answered by
GPT-4o mini
To find the \( y \)-value that corresponds to the \( x \)-value of \( -1 \) for the function \( f(x) = 2x^2 - 2x \), we can substitute \( x = -1 \) into the function:
\[
f(-1) = 2(-1)^2 - 2(-1)
\]
Calculating this step-by-step:
1. Calculate \( (-1)^2 \):
\[
(-1)^2 = 1
\]
2. Substitute back into the function:
\[
f(-1) = 2(1) - 2(-1)
\]
\[
= 2 - (-2)
\]
\[
= 2 + 2
\]
\[
= 4
\]
Therefore, the \( y \)-value that corresponds to the \( x \)-value of \( -1 \) is:
\[
\boxed{4}
\]
\[
f(-1) = 2(-1)^2 - 2(-1)
\]
Calculating this step-by-step:
1. Calculate \( (-1)^2 \):
\[
(-1)^2 = 1
\]
2. Substitute back into the function:
\[
f(-1) = 2(1) - 2(-1)
\]
\[
= 2 - (-2)
\]
\[
= 2 + 2
\]
\[
= 4
\]
Therefore, the \( y \)-value that corresponds to the \( x \)-value of \( -1 \) is:
\[
\boxed{4}
\]
Answered by
GPT-4o mini
To determine if the x-intercepts of the function \( f(x) = 2x^2 - 2x \) are \( (0,0) \) and \( (1,0) \), we need to find the x-intercepts by setting \( f(x) = 0 \):
\[
2x^2 - 2x = 0
\]
We can factor out \( 2x \):
\[
2x(x - 1) = 0
\]
Setting each factor to zero gives us the potential x-intercepts:
1. \( 2x = 0 \) implies \( x = 0 \)
2. \( x - 1 = 0 \) implies \( x = 1 \)
Thus, the x-intercepts are:
- \( (0, 0) \)
- \( (1, 0) \)
Since both of these points are indeed the x-intercepts of the function, the statement is **true**.
Therefore, the answer is:
\[
\text{Type 1 for true.}
\]
So, the answer you should type is **1**.
\[
2x^2 - 2x = 0
\]
We can factor out \( 2x \):
\[
2x(x - 1) = 0
\]
Setting each factor to zero gives us the potential x-intercepts:
1. \( 2x = 0 \) implies \( x = 0 \)
2. \( x - 1 = 0 \) implies \( x = 1 \)
Thus, the x-intercepts are:
- \( (0, 0) \)
- \( (1, 0) \)
Since both of these points are indeed the x-intercepts of the function, the statement is **true**.
Therefore, the answer is:
\[
\text{Type 1 for true.}
\]
So, the answer you should type is **1**.
Answered by
GPT-4o mini
To find the amount spent on marketing that will achieve the maximum profit for the company, we can use the vertex formula for a quadratic equation in the form \( P = ax^2 + bx + c \). In this case, the coefficients from the profit equation \( P = -5x^2 + 1000x + 5000 \) are:
- \( a = -5 \)
- \( b = 1000 \)
- \( c = 5000 \)
The x-value that maximizes the profit can be found using the vertex formula:
\[
x = -\frac{b}{2a}
\]
Now, substituting in the values for \( b \) and \( a \):
\[
x = -\frac{1000}{2 \times -5} = \frac{1000}{10} = 100
\]
This means that to achieve the maximum profit, the company should spend:
\[
\boxed{100} \text{ thousand dollars}
\]
So, the final answer is \( 100,000 \) dollars.
- \( a = -5 \)
- \( b = 1000 \)
- \( c = 5000 \)
The x-value that maximizes the profit can be found using the vertex formula:
\[
x = -\frac{b}{2a}
\]
Now, substituting in the values for \( b \) and \( a \):
\[
x = -\frac{1000}{2 \times -5} = \frac{1000}{10} = 100
\]
This means that to achieve the maximum profit, the company should spend:
\[
\boxed{100} \text{ thousand dollars}
\]
So, the final answer is \( 100,000 \) dollars.
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