Q. 46 - 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 $t=1$ seconds, with $h=0.5, h=0.25, h=-0.5 and h=-0.2$

Step-by-Step Solution

Answer

Ans:   When $h=0.5$ instantaneous velocity is: $-40$

          When $h=0.25$ instantaneous velocity is: $-36$

          When $h=-0.5$ instantaneous velocity is: $-24$

          When $h=-0.2$ instantaneous velocity is: $-28.8$

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,  

   $h=0.5, h=0.25, h=-0.5 and h=-0.2$

Step 2. When the ball has just dropped the value of t = 1 .

To find the instantaneous velocity for $h=0.5$ follows the steps:

Now, 

   $f\left(t\right)=f\left(1\right)=384$

And

    $f(t+h)=f(1.5)=364$

Therefore the instantaneous velocity is: 

     $\begin{array}{r}\frac{f(t+h)-f\left(t\right)}{h}=\frac{364-384}{0.5}\\ =\frac{-20}{0.5}\\ =-40\end{array}$

Step 3. To find the instantaneous velocity for h = 0 . 25 follows the steps:

  $\begin{array}{r}f\left(t\right)=f\left(1\right)=384\\ \end{array}$

And

    $f(t+h)=f(1.25)=375$

Therefore the instantaneous velocity is: 

    $\begin{array}{r}\frac{f(t+h)-f\left(t\right)}{h}=\frac{375-384}{0.25}\\ =\frac{-9}{0.25}\\ =-36\end{array}$

Step 4. To find the instantaneous velocity for h = - 0 . 5 follows the steps:

And

    $f(t+h)=f(0.5)=396$

Therefore the instantaneous velocity is: 

     $\begin{array}{r}\frac{f(t+h)-f\left(t\right)}{h}=\frac{396-384}{-0.5}\\ =\frac{12}{-0.5}\\ =-24\end{array}$

Step 5. To find the instantaneous velocity for h = - 0 . 2 follows the steps:

   $f\left(t\right)=f\left(1\right)=384$

And

    $f(t+h)=f(0.8)=389.76$   $f(t+h)=f(0.8)=389.76$

Therefore the instantaneous velocity is: 

   $\begin{array}{r}\frac{f(t+h)-f\left(t\right)}{h}=\frac{389.76-384}{-0.2}\\ =\frac{5.76}{-0.2}\\ =-28.8\end{array}$