Equation:
Q kπ (d^2)(Th - Tc)
----- = -------------------------
Δt 4L
Given that all quantitiies except d and Δt are constant you can write:
1
------- = C (d^2) => Δt = D / (d^2) (where D = 1/C)
Δt
If d is made three times larger, does the equation predict that Δt will get larger or smaller? By what factor will Δt change, if at all?
Δt = D / (d^2)
Now d' = 3d => Δt' = D / (3d)^2 = D / [9 d^2] = D/(d^2) * 1/9 = Δt / 9
Δt' = Δt / 9 => It predicst that Δt will become smaller and the factor of change is 1/9
What pattern of proportionality of Δt to d does the equation predict
The equation predicts that Δt is inversely related to d^2.
To display this proportionality as a straight line on a graph, what quantities should you plot on the horizontal and vertical axes?
You should plot Δt on the vertical axis and 1/(d^2) on the horizontal axis.
What expression represents the theoretical slope of this graph? (Use k, L, Q, Th, and Tc as necessary.)
The theoretical slope of that graph is what I called D, which I am going to calculate now step by step in terms of k, L, Q, Th and Tc:
Q kπ (d^2)(Th - Tc)
----- = ------------------------- =>
Δt 4L
4QL 1
Δt = -------------- * ----
kπ(Th-Tc) d^2
=> the slope is 4QL / [kπ (Th - Tc) ]