Question

In a small town, the use of coal (W in tons/per

month) isinversely proportional to the averagemonthly
temperature
(T in degrees). The amount of coal used at an average temperature
of 25 degrees is 100Tons.
(i) Write theequation describing the use of coal
W(T).
(ii)Find theamount of coal used during a month
where T = 5 degrees.
(iii)Show thatW(T) is decreasing, and find the
gradient of W(T) at T =10.
(iv) What happensto the coal use W, as the average
monthly Temperature T gets closerto T = 0. Is this formulation is
then practical ?

Answer 1

i)W = 2500 / T

ii) W = 500 Tons

iii) grad W(10°) = - 25î

iv) The formulation is not practical

Explanation:

i) Write an equation describing the use of coal

As use of coal is inversely proportional to the average monthly temperature

if W is use of coal in tons/per month then

W(t) = k / T where k is a constant of proportionality and T is the average temperature in degrees. We have to determine k from given conditions

k = ?? we know that when T = 25° W = 100 tons the by subtitution

W = k/T 100 = k /25 k = 2500 Tons*degree

Then final equation is:

W = 2500 / T

ii) Find the amount of coal when T = 5 degrees

W = 2500 / 5

W = 500 Tons

iii)

The inverse proportionality implies that W will decrease as T increase.

The vector gradient of W function is:

grad W = DW(t)/dt î

grad W = - 2500/T² î

Wich agrees with the fact that W is decreasing.

And when T = 100°

grad W(10°) = - 2500/ 100 î ⇒ grad W(10°) = - 25î

iv) When T = 0 The quantity of coal tends to infinite and the previous formulation is not practical