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 - 15t². 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 of the form $y = ax^{2}+bx + c$, where $a=-3$, $b = 750$, and $c = 0$.

Step2: Find the time $t$ for maximum height

For a quadratic function $y = ax^{2}+bx + c$, the $x$ - coordinate (in our case, the time $t$) of the vertex is given by $t=-\frac{b}{2a}$. Substitute $a=-3$ and $b = 750$ into the formula:
\[t=-\frac{750}{2\times(-3)}=\frac{750}{6}=125\]

Step3: Find the maximum height

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

Answer:

Time to reach maximum height: $t = 125.00$
Maximum height: $h = 46875.00$