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 36 ft/s. its height in feet after t seconds is given by y = 36t - 24t^2. a.) find the average velocity for the time period beginning when t0 = 2 second and lasting for the given time. t = .01 sec -60.24, t = .005 sec -60.12, t = .002 sec -60.048, t = .001 sec 60.024. b.) estimate the instantaneous velocity when t = 2. answer -60. note: for the above answers, you may have to enter 6 or 7 significant digits if you are using a calculator.

Explanation:

Step1: Recall the average - velocity formula

The average - velocity formula over the interval $[t_0,t_1]$ is $v_{avg}=\frac{y(t_1)-y(t_0)}{t_1 - t_0}$. Here, $t_0 = 2$ and we are given different values of $t_1$. The height function is $y(t)=36t - 24t^{2}$.

Step2: Calculate $y(2)$

$y(2)=36\times2-24\times2^{2}=72 - 96=- 24$.

Step3: Calculate average velocity for each $t_1$ value

For $t_1 = 2.01$

$y(2.01)=36\times2.01-24\times(2.01)^{2}=72.36-24\times4.0401=72.36 - 96.9624=-24.6024$.
$v_{avg}=\frac{y(2.01)-y(2)}{2.01 - 2}=\frac{-24.6024+24}{0.01}=\frac{-0.6024}{0.01}=-60.24$.

For $t_1 = 2.005$

$y(2.005)=36\times2.005-24\times(2.005)^{2}=72.18-24\times4.020025=72.18 - 96.4806=-24.3006$.
$v_{avg}=\frac{y(2.005)-y(2)}{2.005 - 2}=\frac{-24.3006 + 24}{0.005}=\frac{-0.3006}{0.005}=-60.12$.

For $t_1 = 2.002$

$y(2.002)=36\times2.002-24\times(2.002)^{2}=72.072-24\times4.008004=72.072 - 96.192096=-24.120096$.
$v_{avg}=\frac{y(2.002)-y(2)}{2.002 - 2}=\frac{-24.120096+24}{0.002}=\frac{-0.120096}{0.002}=-60.048$.

For $t_1 = 2.001$

$y(2.001)=36\times2.001-24\times(2.001)^{2}=72.036-24\times4.004001=72.036 - 96.096024=-24.060024$.
$v_{avg}=\frac{y(2.001)-y(2)}{2.001 - 2}=\frac{-24.060024 + 24}{0.001}=\frac{-0.060024}{0.001}=-60.024$.

To estimate the instantaneous velocity at $t = 2$, we observe the trend of these average - velocity values as $t_1$ gets closer to $2$.

Answer:

a) For $t_1 = 2.01$: $-60.24$ ft/s; for $t_1 = 2.005$: $-60.12$ ft/s; for $t_1 = 2.002$: $-60.048$ ft/s; for $t_1 = 2.001$: $-60.024$ ft/s.
b) The instantaneous velocity at $t = 2$ is approximately $-60$ ft/s.