David K. answered • 04/20/22
Expert, Patient Calculus Tutor with 5000+ Hours Tutoring Experience
Great question Lissie. To approximate the amount of water used during the shower, we need to add up the amounts of water used during all the phases of the shower, keeping in mind that the rate of water usage changes. Since our y-values give the rate at which water is being used and our x-values give the time, the total amount of water used will be given by the area between the x-axis and the graph that show sthe rate of water flow.
The trapezoidal rule says that we can approximate the area under a curve by breaking it up into segments, each of which is a trapezoid, and then adding up the areas of those trapezoids. The area formula for a trapezoid is:
A = h * (b1 + b2)/2
where A is the are of the trapezoid, h is the height of the trapezoid, and b1 and b2 are the lengths of its bases.
The first trapezoid we will use for this problem covers the region of the graph between the vertical lines x = 0 and x = 1. We can imagine the height of this trapezoid as being 1 (the distance from x = 0 to x = 1 along the x-axis), and the lengths of its two bases as being the values of the function at x = 0 and x = 1.
The area calculation woud look like this:
A = 1 * (0.3 + 1.6)/2 = 0.95
Our next trapezoid goes from x = 1 to x = 2. Again, it will have a height of 1 and base lengths equal to f(1) and f(2), so we can find its area as follows:
A = 1 * (1.6 + 2.0)/2 = 1.8
Continuing this pattern for all other trapezoids, we will add up the areas of our 8 trapezoids, and find an approximation of 13.9 gallons for water usage. I hope this helps! Let me know if any questions and we can discuss further.
