Q. 47 - Calculus

Question

A bowling ball dropped from a height of $400$ feet will be $s\left(t\right)=400-16t^2$ feet from the ground after $t$ seconds. Use a sequence of average velocities to estimate the instantaneous velocities described below:

     After $t=2$ seconds, with $h=0.1, h=0.01 h=-0.1 and h=-0.01$

Step-by-Step Solution

Answer

Ans:  When $h=0.1$ instantaneous velocity is: $-65.4$

         When $h=0.01$ instantaneous velocity is: $-65$

         When $h=-0.1$ instantaneous velocity is: $-62.4$

         When $h=-0.01$ instantaneous velocity is: $-64$

Step 1. Given information.

given,  

     A ball is dropped  from a height of $400$ feet and its distance from the ground after $t$ seconds is,  $s\left(t\right)=400-16t^2$
The objective is to estimate the instantaneous velocities for, 

 $t=2 s, h=0.1, h=0.01, h=-0.1 and h=-0.01$

Step 2. To find the instantaneous velocity for h = 0 . 1 follows the steps:

    $f\left(t\right)=f\left(2\right)=400-16{\left(2\right)}^2=336$

And

     $f(t+h)=f(2.1)=329.44$

Therefore the instantaneous velocity is: 

$\begin{array}{r}\frac{f(t+h)-f\left(t\right)}{h}=\frac{329.44-336}{0.1}\\ =\frac{-6.54}{0.1}\\ =-65.4\end{array}$

Step 3. To find the instantaneous velocity for h = 0 . 01 follows the steps:

    $f\left(t\right)=f\left(2\right)=336$

And

    $f(t+h)=f(2.01)=335.35$

Therefore the instantaneous velocity is: 

$\begin{array}{r}\frac{f(t+h)-f\left(t\right)}{h}=\frac{335.35-336}{0.01}\\ =\frac{-0.65}{0.01}\\ =-65\end{array}$

Step 4. To find the instantaneous velocity for h = - 0 . 1 follows the steps:

   $f\left(t\right)=f\left(2\right)=336$

And

    $f(t+h)=f(1.9)=342.24$

Therefore the instantaneous velocity is: 

$\begin{array}{r}\frac{f(t+h)-f\left(t\right)}{h}=\frac{342.24-336}{-0.1}\\ =\frac{6.24}{-0.1}\\ =-62.4\end{array}$

$f(t+h)=f(1.9)=324.24$

Step 5. To find the instantaneous velocity for h = - 0 . 01 follows the steps:

$f\left(t\right)=f\left(2\right)=336$

And,   $f(t+h)=f(1.99)=336.64$

Therefore the instantaneous velocity is: 

$\begin{array}{r}\frac{f(t+h)-f\left(t\right)}{h}=\frac{336.64-336}{-0.2}\\ =\frac{0.64}{-0.01}\\ =-64\end{array}$