Question

a ball is thrown into the air by a baby alien on a planet in the system of alpha centauri with a velocity of 29 ft/s. its height in feet after t seconds is given by y = 29t - 27t². a. find the average velocity for the time period beginning when t = 3 and lasting .01 s: .005 s: .002 s: .001 s: note: for the above answers, you may have to enter 6 or 7 significant digits if you are using a calculator. estimate the instantaneous velocity when t = 3. question help: video message instructor

Explanation:

Step1: Recall the formula for average velocity

The average velocity $v_{avg}=\frac{\Delta y}{\Delta t}$, and the instantaneous velocity $v$ is the derivative of the position - function $y(t)$. Given $y = 29t-27t^{2}$. First, find the derivative using the power rule $\frac{d}{dt}(at^{n})=nat^{n - 1}$.
$y^\prime(t)=v(t)=\frac{d}{dt}(29t - 27t^{2})=29-54t$.

Step2: Calculate the instantaneous velocity at $t = 3$

Substitute $t = 3$ into the velocity - function $v(t)$.
$v(3)=29-54\times3=29 - 162=- 133$ ft/s.

Step3: Calculate average velocity for a small time - interval around $t = 3$

For a small time - interval $[3,3 + h]$:
$\Delta y=y(3 + h)-y(3)$.
$y(3)=29\times3-27\times3^{2}=87 - 243=-156$.
$y(3 + h)=29(3 + h)-27(3 + h)^{2}=87+29h-27(9 + 6h+h^{2})=87+29h - 243-162h-27h^{2}=-156 - 133h-27h^{2}$.
$\Delta y=y(3 + h)-y(3)=(-156 - 133h-27h^{2})-(-156)=-133h-27h^{2}$.
$\Delta t=h$.
$v_{avg}=\frac{\Delta y}{\Delta t}=\frac{-133h-27h^{2}}{h}=-133 - 27h$.

For $h = 0.01$:

$v_{avg}=-133-27\times0.01=-133 - 0.27=-133.27$ ft/s.

For $h = 0.005$:

$v_{avg}=-133-27\times0.005=-133 - 0.135=-133.135$ ft/s.

For $h = 0.002$:

$v_{avg}=-133-27\times0.002=-133 - 0.054=-133.054$ ft/s.

For $h = 0.001$:

$v_{avg}=-133-27\times0.001=-133 - 0.027=-133.027$ ft/s.

Answer:

Instantaneous velocity at $t = 3$: - 133 ft/s
Average velocity for $t\in[3,3.01]$: - 133.27 ft/s
Average velocity for $t\in[3,3.005]$: - 133.135 ft/s
Average velocity for $t\in[3,3.002]$: - 133.054 ft/s
Average velocity for $t\in[3,3.001]$: - 133.027 ft/s