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 and identify the x-intercept(s) and its meaning.

Question

Responses

(0, 1.563); 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.

(0.781, 9.766); The soccer ball will reach a maximum height of 9.766 feet 0.781 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 understand the function \( f(x) = -16x^2 + 25x \), we need to analyze its components:

1. Identify the x-intercepts: The x-intercepts occur when \( f(x) = 0 \). So, we need to solve the equation: \[ -16x^2 + 25x = 0 \] We can factor out \( x \): \[ x(25 - 16x) = 0 \] This gives us two solutions: \[ x = 0 \quad \text{or} \quad 25 - 16x = 0 \Rightarrow 16x = 25 \Rightarrow x = \frac{25}{16} \approx 1.563 \]

   Therefore, the x-intercepts are \( (0, 0) \) and \( (1.563, 0) \).

2. Interpret the x-intercepts: In the context of the soccer ball's motion:

   • The first intercept \( (0, 0) \) indicates the moment the ball is kicked from the ground.
   • The second intercept \( (1.563, 0) \) indicates that the soccer ball will return to the ground approximately 1.563 seconds after it has been kicked.

So, the correct answer based on the x-intercepts is: (1.563, 0); The soccer ball will return to the ground 1.563 seconds after it has been kicked.