Question

If a football player passes a football from 4 feet off the ground with an initial velocity of 6 feet per second, how long will it take the football to hit the ground? Use the equation h = −16t2 + 6t + 4. Round your answer to the nearest hundredth.

0.72
0.65
0.35
0.27

Answer 1

To find the time it takes for the football to hit the ground, we need to solve the equation \( h = -16t^2 + 6t + 4 \) for \( h = 0 \) (when the football hits the ground).

We set up the equation:

\[
-16t^2 + 6t + 4 = 0
\]

Now, we can use the quadratic formula, which is:

\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In our equation, \( a = -16 \), \( b = 6 \), and \( c = 4 \).

First, we calculate the discriminant:

\[
b^2 - 4ac = 6^2 - 4(-16)(4) = 36 + 256 = 292
\]

Now we apply the quadratic formula:

\[
t = \frac{-6 \pm \sqrt{292}}{2(-16)}
\]

Calculating \( \sqrt{292} \):

\[
\sqrt{292} \approx 17.09
\]

Now substituting back into the quadratic formula:

\[
t = \frac{-6 \pm 17.09}{-32}
\]

This gives us two possible solutions for \( t \):

1.

\[
t = \frac{-6 + 17.09}{-32} = \frac{11.09}{-32} \approx -0.346
\]

(This value is negative and thus not a valid solution since time cannot be negative.)

2.

\[
t = \frac{-6 - 17.09}{-32} = \frac{-23.09}{-32} \approx 0.722
\]

We round \( 0.722 \) to the nearest hundredth:

\[
t \approx 0.72
\]

Thus, the time it takes for the football to hit the ground is **0.72 seconds**.