How do I work this problem?

Question

Trash is delivered to a dump-site between the hours of midnight (t = 0) and 6 a.m. (t = 6). Selected values of the rate of delivery are shown in the table below, with t measured in hours past midnight, and R(t) measured in tons per hour. Use 4 trapezoids to estimate the total amount of trash rounded up to the nearest ton that is delivered between midnight and 6 a.m.

Answer 1

You have not included the table of values showing the delivery rates, but I can give you the steps and let you apply those to the specific data given with the question:

We are going to calculate the areas of 4 separate trapezoids and sum those areas. This sum will be an approximation of the area under the curve R(t). We interpret this area as the total amount of trash delivered between t = 0 and t = 6.

These trapezoids are adjacent to one another, with their parallel sides (bases) oriented vertically, with each adjacent pair sharing one base. We will use the formula for area of a trapezoid: A_trap = (b₁ + b₂)/2 · h

Note that because of their orientation, the height, h, will be horizontal, which we calculate by subtracting consecutive time values.

The table should give 5 distinct time values and their corresponding rates. I assume that the first time value given is t = 0 and the last is t = 6. I will refer to the other 3 time values as t₁ , t₂ , and t₃.

A = (R(0) + R(t₁))/2 · (t₁) + (R(t₁) + R(t₂))/2 · (t₂ - t₁) + (R(t₂) + R(t₃))/2 · (t₃ - t₂) + (R(t₁) + R(6))/2 · (6 - t₃)