Question

Use the quadratic formula to calculate the time it will take for a t-shirt shot outside the stands to hit the ground given the function h(t)=−16t2+32t+5 . Round to the nearest hundredth, if necessary. Reference Sheet - Quadratic Formula (1 point) A t-shirt will hit the ground at about seconds.

Answer 1

To find the time it takes for the t-shirt to hit the ground using the function \( h(t) = -16t^2 + 32t + 5 \), we need to determine when the height \( h(t) \) equals zero.

This sets up the equation:

\[
-16t^2 + 32t + 5 = 0
\]

We can apply the quadratic formula, which is given by:

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

In our equation, the coefficients are as follows:

- \( a = -16 \)
- \( b = 32 \)
- \( c = 5 \)

Let's calculate the discriminant (\( b^2 - 4ac \)) first:

\[
b^2 = 32^2 = 1024
\]
\[
4ac = 4 \times (-16) \times 5 = -320
\]
\[
b^2 - 4ac = 1024 - (-320) = 1024 + 320 = 1344
\]

Now we can apply the quadratic formula:

\[
t = \frac{-32 \pm \sqrt{1344}}{2 \times -16}
\]

Calculating \( \sqrt{1344} \):

\[
\sqrt{1344} \approx 36.66
\]

Now, substituting back into the formula:

\[
t = \frac{-32 \pm 36.66}{-32}
\]

This will yield two possible solutions for \( t \):

1. \( t = \frac{-32 + 36.66}{-32} = \frac{4.66}{-32} \)
\[
t \approx -0.1456 \quad (\text{not a valid time since it's negative})
\]

2. \( t = \frac{-32 - 36.66}{-32} = \frac{-68.66}{-32} \)
\[
t \approx 2.144375
\]

Rounding to the nearest hundredth gives:

\[
t \approx 2.14 \text{ seconds}
\]

Thus, the t-shirt will hit the ground at approximately **2.14 seconds**.