Question

question 23 of 30
a kicker kicks a football toward the opponents goal line during a game. the ball begins its flight with an initial velocity of 100 feet per second when it is kicked at a height of 3 feet by the kicker. to the nearest foot, what maximum height will the ball reach?
you may use the function:
h(t)=-16t^{2}+100t + 3
a. 100 feet
b. 129 feet
c. the answer cannot be determined.
d. 159 feet

Explanation:

Step1: Find the time of maximum height

The function $h(t)=-16t^{2}+100t + 3$ is a quadratic function in the form $y = ax^{2}+bx + c$ with $a=-16$, $b = 100$, $c = 3$. The time $t$ at which the ball reaches its maximum height is given by the formula $t=-\frac{b}{2a}$. So, $t=-\frac{100}{2\times(-16)}=\frac{100}{32}=\frac{25}{8}$ seconds.

Step2: Calculate the maximum height

Substitute $t = \frac{25}{8}$ into the height - function $h(t)$.
\[

$$\begin{align*} h(\frac{25}{8})&=-16\times(\frac{25}{8})^{2}+100\times\frac{25}{8}+3\\ &=-16\times\frac{625}{64}+\frac{2500}{8}+3\\ &=-\frac{625}{4}+\frac{2500}{8}+3\\ &=-\frac{1250}{8}+\frac{2500}{8}+3\\ &=\frac{- 1250 + 2500}{8}+3\\ &=\frac{1250}{8}+3\\ &=\frac{1250+24}{8}\\ &=\frac{1274}{8}\\ & = 159.25\approx159 \end{align*}$$

\]

Answer:

D. 159 feet