Calculus help...?

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 34 ft/s. Its height in feet after t seconds is given by:

y=34t - 22t^2 (t is squared)

A. Find the average velocity for the time period beginning when t=3 and lasting
.01 s:
.005 s:
.002 s:
.001 s:


B. Estimate the instanteneous velocity when t=3.

Answer 1

Hi,
Unfortunately, there is something wrong with this problem—a decimal point missing, a mistyped number, etc.
For example, if we set y = 0 (when it was launched and when it hit the ground or whatever surface they have on Alpha Centauri).
34t -22t² = 0
t(34-22t) = 0
t =0
34-22t = 0
-22t = -34
t = 1.545
So, the thing hit the ground at t = 1.545 s, and it’s just sitting there at t=3 unless the surface is very pliable on this planet and we have no information about that.
But, let’s just modify that problem so that it works and then see how we would do it.
Let’s say we let the periods begin at t = 1.0 sec.
Then our approach would be:
1) Find the values of “y” at t = 1.0 and 1.01 and find the difference.
2) Now, since v = d/t, we divide that difference by the elapsed time, .01.
Let’s do it:

For y= 1.0
y=34t -22t²
=34(1)-22(1)²
=34 -22
=12 ft

For y = 1.01
Y= 34(1.01) -22(1.01)²
= 11.8978 ( Incidentally, this is more indication of a problem with your numbers. Notice that the object is already on its way downward at t = 1.0 sec since it’s height has decreased from 12 ft to 11.89….)
Now to calculate the velocity.
v = 12-11.8978/.01
= 10.22 ft/s

I won’t work through any more numbers since the numbers have a problem, but I’ll outline the velocity for part b.
There are a couple approaches you could take, and we’d need to see the context of your previous problems to see which one they want.
1) After you find the velocity at t = 1.001, just assume it’s somewhat smaller than that and make an estimate.
2) Another more accurate way is to find the slope of the velocity curve. M = (v2-v1)/(t2-t1).
Multiply that by the elapsed time, .001 and subtract it from the velocity for
t=1.001
Hope this helps. If you need more help, repost after you’ve studied your numbers.
FE