Q. 45 - 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:

       When the bowling ball is first dropped, with $h=0.5,$$h=0.25, and h=0.1$

Step-by-Step Solution

Answer

Ans:  When $h=0.5$ instantaneous velocity is: $-8$

         When $h=0.25$ instantaneous velocity is: $-4$

         When $h=0.1$ instantaneous velocity is: $-1.6$

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, 

            $h=0.5, h=0.25, h=0.1$

Step 2. When the ball has just dropped the value of c = 0 .

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

Now, 

   $$\begin{gathered} f\left(c\right)=f\left(0\right) \\ =400 \end{gathered}$$

And

    $f(c+h)=f(0.5)=396$

therefore the instantaneous velocity is:

           $\frac{f(c+h)-f\left(c\right)}{h}=\frac{396-400}{0.5}=-8$

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

$f\left(c\right)=f\left(0\right)=400$

And,

  $f(c+h)=f(0.25)=399$

therefore the instantaneous velocity is:

  $\frac{f(c+h)-f\left(c\right)}{h}=\frac{399-400}{0.25}=-4$

$f(c+h)=f(0.25)=399$

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

$f\left(c\right)=f\left(0\right)=400$

And,

   $f(c+h)=f(0.1)=399.84$

therefore the instantaneous velocity is:

  $\frac{f(c+h)-f\left(c\right)}{h}=\frac{399.84-400}{0.1}=-1.6$