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 $h=0.5,h=0.25,h=-0.5andh=-0.2$

Short 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-by-step solution

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,

$h=0.5,h=0.25,h=-0.5andh=-0.2$

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:

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