Question

Use the quadratic formula on the function h=−16t2+80t+3
to calculate the time it will take for the baseball to hit the ground. Round to the nearest hundredth, if necessary.

Reference Sheet - Quadratic Formula

(1 point)
The baseball will hit the ground at about
seconds.

Answer 1

To find the time it takes for the baseball to hit the ground, we need to set the height \( h \) to zero and use the quadratic formula. The equation given is:

\[
h = -16t^2 + 80t + 3
\]

Setting \( h = 0 \) gives us the equation:

\[
-16t^2 + 80t + 3 = 0
\]

In standard form \( at^2 + bt + c = 0 \), we have:

- \( a = -16 \)
- \( b = 80 \)
- \( c = 3 \)

The quadratic formula is given by:

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

Now, we will substitute the values of \( a \), \( b \), and \( c \) into the formula.

First, calculate the discriminant \( b^2 - 4ac \):

\[
b^2 = 80^2 = 6400
\]
\[
4ac = 4 \cdot (-16) \cdot 3 = -192
\]
\[
b^2 - 4ac = 6400 - (-192) = 6400 + 192 = 6592
\]

Now, we calculate \( t \):

\[
t = \frac{-80 \pm \sqrt{6592}}{2 \cdot -16}
\]

Next, calculate \( \sqrt{6592} \):

\[
\sqrt{6592} \approx 81.1 \quad (\text{approximately})
\]

Now, substitute this back into the formula:

\[
t = \frac{-80 \pm 81.1}{-32}
\]

We have two possible solutions for \( t \):

1. Using the positive root:
\[
t = \frac{-80 + 81.1}{-32} = \frac{1.1}{-32} \approx -0.034375 \quad (\text{not a physical solution, as time can't be negative})
\]

2. Using the negative root:
\[
t = \frac{-80 - 81.1}{-32} = \frac{-161.1}{-32} \approx 5.03
\]

Therefore, the baseball will hit the ground at about:

\[
\boxed{5.03} \text{ seconds}
\]