Q. 2 - Calculus

Question

A limit representing an instantaneous rate of change: After t seconds, a bowling ball dropped from $350$feet has height $h\left(t\right)=350-16t^2,$measured in feet.

Write down a formula for the average rate of change of the height of the bowling ball from time $t=3$to time $t=3+h,$assuming that $h>0.$The only letter in your formula should be h.

Step-by-Step Solution

Answer

The formula for the average rate of change of the height of the bowling ball is $r=-16h-96.$

Step 1. Given information.

The given function for the height is $h\left(t\right)=350-16t^2.$

Given time is $t=3 \mathrm{to} t=3+h.$

Step 2. The average rate of change in the height

 The average rate of change of the height of the bowling ball from $t=3$to $t=3+h$seconds is as follows.

$$\begin{gathered} r=\frac{h(h+3)-h\left(3\right)}{(t+3)-3} \\ r=\frac{\left[350-16{(h+3)}^2\right]-\left[350-16\left(3^2\right)\right]}{h+3-3} \\ r=\frac{350-16h^2-96h-144-350+144}{h} \\ r=\frac{-16h^2-96h}{h} \\ r=-16h-96 \end{gathered}$$

so the formula for the average rate of change of the height is$r=-16h-96$