The table of values below shows the rate of water consumption in gallons per hour at selected time intervals from t = 0 to t = 12.

Question

The table of values below shows the rate of water consumption in gallons per hour at selected time intervals from t = 0 to t = 12.

Using a Trapezoidal sum with 5 subintervals, estimate the total amount of water consumed in that time interval.

x 0 2 5 7 11 12

f(x)	5.7	5.0	2.0	1.2	0.6	0.4

28.5

35.76 

56.88

None of these

Answer 1

The total amount of water is the integral of the water consumption rate over the time in question (0 - 12) or, in other words, the area under the curve. Here is a graph showing the function values and sub-intervals given.

The area of a trapezoid is 1/2(b₁ + b₂)(h). In these trapezoids, the height (h) is the x distance of each interval and the b₁ and b₂ are the function values (the f(x) or y-values) at each side of the interval.

The area of Trapezoid 1 is 1/2(5.7 + 5)(2) = 10.7

The area of Trapezoid 2 is 1/2(5 + 2)(3) = 10.5

The area of Trapezoid 3 is 1/2(2 + 1.2)(2) = 3.2

The area of Trapezoid 4 is 1/2(1.2 + 0.6)(4) = 3.6

The area of Trapezoid 5 is 1/2(0.6 + 0.4)(1) = 0.5

So the water consumed is 10.7 + 10.5 + 3.2 + 3.6 + 0.5 = 28.5 gallons