Question
A basketball is shot from the free-throw line. The function h(t)=−16t^2+40t+6
gives the height h(t)
of the basketball in feet, t
seconds after it is released.
What is the value of h(1)
? Response area
What does h(2.5)
represent in the context of this problem? Response area
What does h(−1)
represent in the context of this problem?
gives the height h(t)
of the basketball in feet, t
seconds after it is released.
What is the value of h(1)
? Response area
What does h(2.5)
represent in the context of this problem? Response area
What does h(−1)
represent in the context of this problem?
Answers
Answered by
GPT-4o mini
To find the value of \( h(1) \), we substitute \( t = 1 \) into the function \( h(t) = -16t^2 + 40t + 6 \):
\[
h(1) = -16(1)^2 + 40(1) + 6
\]
\[
h(1) = -16(1) + 40 + 6
\]
\[
h(1) = -16 + 40 + 6
\]
\[
h(1) = 30
\]
So, \( h(1) = 30 \) feet.
---
To understand what \( h(2.5) \) represents in the context of this problem, we evaluate \( h(2.5) \):
\[
h(2.5) = -16(2.5)^2 + 40(2.5) + 6
\]
\[
= -16(6.25) + 100 + 6
\]
\[
= -100 + 100 + 6
\]
\[
= 6
\]
In the context of this problem, \( h(2.5) \) represents the height of the basketball in feet at \( 2.5 \) seconds after it is released. Specifically, when the basketball is \( 2.5 \) seconds into its flight, it is \( 6 \) feet above the ground.
---
Next, we need to evaluate \( h(-1) \):
\[
h(-1) = -16(-1)^2 + 40(-1) + 6
\]
\[
= -16(1) - 40 + 6
\]
\[
= -16 - 40 + 6
\]
\[
= -50
\]
In the context of this problem, \( h(-1) \) represents the height of the basketball in feet at \( -1 \) second, which doesn't have a physical meaning in this scenario since it refers to a time before the basketball was released. Hence, it suggests a height of \(-50\) feet, indicating that a negative height is not physically meaningful within this context; it simply indicates a point in time that is not relevant for this event.
\[
h(1) = -16(1)^2 + 40(1) + 6
\]
\[
h(1) = -16(1) + 40 + 6
\]
\[
h(1) = -16 + 40 + 6
\]
\[
h(1) = 30
\]
So, \( h(1) = 30 \) feet.
---
To understand what \( h(2.5) \) represents in the context of this problem, we evaluate \( h(2.5) \):
\[
h(2.5) = -16(2.5)^2 + 40(2.5) + 6
\]
\[
= -16(6.25) + 100 + 6
\]
\[
= -100 + 100 + 6
\]
\[
= 6
\]
In the context of this problem, \( h(2.5) \) represents the height of the basketball in feet at \( 2.5 \) seconds after it is released. Specifically, when the basketball is \( 2.5 \) seconds into its flight, it is \( 6 \) feet above the ground.
---
Next, we need to evaluate \( h(-1) \):
\[
h(-1) = -16(-1)^2 + 40(-1) + 6
\]
\[
= -16(1) - 40 + 6
\]
\[
= -16 - 40 + 6
\]
\[
= -50
\]
In the context of this problem, \( h(-1) \) represents the height of the basketball in feet at \( -1 \) second, which doesn't have a physical meaning in this scenario since it refers to a time before the basketball was released. Hence, it suggests a height of \(-50\) feet, indicating that a negative height is not physically meaningful within this context; it simply indicates a point in time that is not relevant for this event.
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