Question

a ball is thrown up on the surface of a moon. its height above the lunar surface (in feet) after t seconds is given by the formula h = 750t - \frac{15}{5}t^{2}. find the time that the ball reaches its maximum height. answer = (round to two decimal places as needed.) find the maximal height attained by the ball answer = (round to two decimal places as needed.)

Explanation:

Step1: Identify the function type

The height function $h(t)=750t - \frac{15}{5}t^{2}=750t - 3t^{2}$ is a quadratic function in the form $y = ax^{2}+bx + c$ where $a=- 3$, $b = 750$ and $c = 0$.

Step2: Find the time for maximum height

For a quadratic function $y=ax^{2}+bx + c$, the vertex (where maximum or minimum occurs) has $x$ - coordinate (in our case $t$ - coordinate) given by $t=-\frac{b}{2a}$. Substituting $a=-3$ and $b = 750$ into the formula, we get $t=-\frac{750}{2\times(-3)}$.
$t=\frac{750}{6}=125.00$

Step3: Find the maximum height

Substitute $t = 125$ into the height function $h(t)=750t-3t^{2}$.
$h(125)=750\times125-3\times(125)^{2}$
$h(125)=93750-3\times15625$
$h(125)=93750 - 46875$
$h(125)=46875.00$

Answer:

Find the time that the ball reaches its maximum height: $125.00$
Find the maximal height attained by the ball: $46875.00$