A Simple Word Problem on Finding an Integral of an Expression/Temperature vs Time problem

Here is a Multi choice word problem on Temperature and Time to Evaluate the temperature by Integration. Also converting the value into degrees. You can find similar set of questions at TutorVista Blogs.

Topic : Integration.

Calculation in the solution will help you to identify correct choice and get reasons for, why remaining choices are incorrect.

Question : An observer measures the outside temperature every hour from noon until midnight, recording the temperatures in the following table
Time Temp
N -----63
1 ----- 65
2 ----- 66
3 ----- 68
4 ----- 70
5 ----- 69
6 ----- 68
7 ----- 68
8 ----- 65
9 ----- 64
10 ---- 62
11 ---- 58
M ----- 55

Then the average temperature for the 12-hour period is

a. 71
b. 76
c. 65
d. 66

Solution :
Choice c is correct.

We are looking for the average value of a continuous function (temp.) for which we know values at discrete times that are one unit apart.
We need to find
Average (f) = 1/(b-a)_a∫^bf(x)dx
Without having a formula for f(x)
The integral, however can be approximated by the Trapezoidal Rule, taking h=1

T = h/2(y₀ + 2y₁ + 2y₂ + ..... + 2y₁₁ + 2y₁₂)

= 1/2 (63 + 2*65 + 2*66 + ...... + 2*58 + 2*55)

= 782

average (f)approximately 1/(b-a)T = 1/12*782 = 65.17
Therefore, the average temperature =65 degrees

Choice a is incorrect because of the error in considering the interval between noon to midnight as 13 hours instead of 12 hours.

Choice b is incorrect because of the error in considering the interval between noon to midnight as 11 hours instead of 12 hours.

Choice d is incorrect because of error in calculating the trapezoidal approximation T as 792 instead of 782.

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