Stefan’s Law states that a body emit radiant energy to a sink at absolute zero
according to the formula R = kT4, where R is the measure of the rate of emission of
the rate of energy per square unit of area, T is the measure of the Kelvin temperature
of the surface and k is a constant.
a. Find the average rate of change of R with respect to T as T increases from 200 to
300.
b. The instantaneous rate of change of R with respect to T when T is 200.
I assume you mean R(T) = kT^4
(a) as always, this would be
(R(300)-R(200))/(300-200) = 65*10^6 k
(b) dR/dT = 4kT^3 = 16*10^8 k
To find the average rate of change of R with respect to T, we need to calculate the change in R divided by the change in T. Given the formula R = kT^4, we can differentiate this equation with respect to T to find the rate of change.
a. Average rate of change of R with respect to T:
To find the average rate of change of R with respect to T as T increases from 200 to 300, we differentiate the equation R = kT^4 with respect to T. The derivative of R with respect to T is given by:
dR/dT = 4kT^3
Now we can substitute the values of T into this expression to find the average rate of change. When T increases from 200 to 300, the change in T is 300 - 200 = 100. Substituting these values, we get:
Average rate of change = [R(300) - R(200)] / (300 - 200)
= [k(300)^4 - k(200)^4] / 100
= 80k
So, the average rate of change of R with respect to T is 80k.
b. Instantaneous rate of change of R with respect to T:
To find the instantaneous rate of change of R with respect to T when T is 200, we can use the derivative expression obtained in part a:
dR/dT = 4kT^3
We substitute T = 200 into this expression:
Instantaneous rate of change = dR/dT |(T=200)
= 4k(200)^3
= 32,000,000k
So, the instantaneous rate of change of R with respect to T when T is 200 is 32,000,000k.