To determine the times at which the golf ball will be on the ground, we need to find the values of [tex]\( t \)[/tex] for which [tex]\( h(t) = 0 \)[/tex]. The height function provided is:
[tex]\[
h(t) = -16(t-3)^2 + 144
\][/tex]
Step-by-step solution:
1. Set the height function equal to zero:
[tex]\[
0 = -16(t-3)^2 + 144
\][/tex]
2. Isolate the quadratic term:
[tex]\[
16(t-3)^2 = 144
\][/tex]
3. Divide both sides by 16:
[tex]\[
(t-3)^2 = \frac{144}{16}
\][/tex]
[tex]\[
(t-3)^2 = 9
\][/tex]
4. Take the square root of both sides:
[tex]\[
t-3 = \pm3
\][/tex]
5. Solve for [tex]\( t \)[/tex] by adding 3 to both solutions:
[tex]\[
t - 3 = 3 \quad \text{or} \quad t - 3 = -3
\][/tex]
[tex]\[
t = 3 + 3 \quad \text{or} \quad t = 3 - 3
\][/tex]
[tex]\[
t = 6 \quad \text{or} \quad t = 0
\][/tex]
Thus, the golf ball will be on the ground at [tex]\( t = 0 \)[/tex] seconds and [tex]\( t = 6 \)[/tex] seconds.
The correct statement is:
- The golf ball will be on the ground at 0 and 6 seconds.
