Question

Laura will kick a soccer ball into the air. The model y=−0.004x2+0.489x represents the average path of Laura’s last 20 soccer ball kicks, where x is the time of the ball’s flight (in seconds) and y is the height reached by the ball (in feet). Using the model, predict the maximum height that Laura will kick the ball in the air. Round to the nearest hundredth.(1 point) Laura will kick the soccer ball about feet in the air.

Answer 1

To find the maximum height using the quadratic model given by the equation \( y = -0.004x^2 + 0.489x \), we need to find the vertex of the parabola represented by this equation. For a quadratic equation in the standard form \( y = ax^2 + bx + c \), the x-coordinate of the vertex can be calculated using the formula:

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

In our equation, \( a = -0.004 \) and \( b = 0.489 \). Plugging these values into the formula gives:

\[
x = -\frac{0.489}{2 \times -0.004} = \frac{0.489}{0.008} = 61.125
\]

Now, we find the maximum height by substituting \( x = 61.125 \) back into the original equation:

\[
y = -0.004(61.125)^2 + 0.489(61.125)
\]

Calculating \( (61.125)^2 \):

\[
(61.125)^2 = 3736.515625
\]

Now substituting into the equation for \( y \):

\[
y = -0.004(3736.515625) + 0.489(61.125)
\]
\[
y = -14.9460625 + 29.872125
\]
\[
y = 14.9260625
\]

Now rounding to the nearest hundredth:

\[
y \approx 14.93
\]

Thus, the maximum height that Laura will kick the soccer ball into the air is approximately **14.93 feet**.