Question

A soccer ball is kicked from the ground at an upward velocity of 25 feet per second. This situation can be represented by the function f(x)=−16x^2+25x . Graph the function. Identify the x-intercept(s) and interpret its meaning.(1 point)
Responses

(0.781,9.766); The soccer ball will reach a maximum height of 9.766 feet 0.781 seconds after being kicked.
left parenthesis 0.781 comma 9.766 right parenthesis ; The soccer ball will reach a maximum height of 9.766 feet 0.781 seconds after being kicked.

(0,1.563); The soccer ball will return to the ground 1.563 seconds after it has been kicked.
left parenthesis 0 comma 1.563 right parenthesis ; The soccer ball will return to the ground 1.563 seconds after it has been kicked.

(1.25,0); The soccer ball will return to the ground 1.25 seconds after it has been kicked.
left parenthesis 1.25 comma 0 right parenthesis ; The soccer ball will return to the ground 1.25 seconds after it has been kicked.

(1.563,0); The soccer ball will return to the ground 1.563 seconds after it has been kicked.

Answer 1

To analyze the situation, let's first understand the function \( f(x) = -16x^2 + 25x \), which represents the height of the soccer ball in feet after \( x \) seconds.

To find the x-intercepts (where the ball hits the ground), we need to set \( f(x) = 0 \):

\[
-16x^2 + 25x = 0
\]

Factoring out \( x \):

\[
x(-16x + 25) = 0
\]

Thus, we have:

1. \( x = 0 \) (when the ball is kicked)
2. \( -16x + 25 = 0 \) leads to \( 16x = 25 \Rightarrow x = \frac{25}{16} = 1.5625 \)

So, the x-intercepts are \( x = 0 \) and \( x = 1.5625 \) seconds.

The meaning of these x-intercepts:
- \( x = 0 \): This indicates the initial moment when the soccer ball is kicked.
- \( x = 1.5625 \): This indicates the moment when the soccer ball returns to the ground.

Thus the x-intercept we care about is approximately \( (1.563, 0) \), which means the soccer ball will return to the ground approximately **1.563 seconds** after it has been kicked.

Based on the given answers, the correct response is:

**(1.563, 0); The soccer ball will return to the ground 1.563 seconds after it has been kicked.**