Question

CALCULUS QUESTION!
Temperature within a greenhouse over 24-hour period is given as T(t)=21+sin(pi t/12)





a) determine the instantaneous rate of change at t=8... I found this by determining the derivative, and found the rate to be 0.85
b) determine the average rate of change of temp, within the greenhouse b/w the hours of 4-12.
c) determine the avg temperature of greenhouse between the hours of 4-12.
d) determine the average daily temp of the greenhouse.

Answer 1

but the thing is i have to use integration.. but i don't understand how i would apply that concept to these questions.

Answer 2

b. Avg. rate of change is found by using the slope formula. Finding the different between the values of the two points, and dividing that by the difference in the x-value.

c. Average value of a closed interval can be calculated like this, if given a function:
\[\frac{\int\limits_{a}^{b} f(x) dx}{b-a}\]

Answer 3

uhm for d, since it's daily so 0-24 can't i just the same avg formula from b

Answer 4

use**

Answer 5

OH, it says 24 hour period lol.
I believe you can still use that formula.

Answer 6

Forget what I said about D earlier lol

Answer 7

Alright! When I do average rate of change, do I use the original formula?

Answer 8

What do you mean original formula?
Average rate of change on a closed interval of a function is found by using the slope formula.

Answer 9

I mean do I use T(t)=21+sin(pit/12) to find the corresponding temperature to the values t=4, and t=12 and then plug that into the slope formula?

Answer 10

Yesh

Answer 11

Okay! When i did that for d, i got 0. lol..

Answer 12

which i don't think is right for the average daily temperature..

Answer 13

I had a feeling it was 0, since there is the sine function there. It's a cyclical thing. When t = 0, you get sin(0) which is 0.
When you plug in t = 24,
sin(24pi/12) = sin(2pi) = 0

Answer 14

That didn't really prove why it's 0, but even if you plug in t = 12, the value is 0.
So, t = 0 to 24 is one cycle of a sine wave, and the average value of that is 0.

Answer 15

Hmmm, alright. Thanks.

Answer 16

:D