A tank initially contains 120 liters of oil. Oil is being pumped out at a rate of $ R(t) $, where $ R(t) $ is measured in gallons per hour, and $ t $ is measured in hours. The table below shows selected values for $ R(t) $.

Using a trapezoidal approximation with three subintervals and the data from the table, estimate the number of gallons of oil remaining in the tank at $ t = 14 $ hours.

| $ t $ (hours) | 2 | 5 | 10 | 14 |
|-----------------|----|----|----|----|
| $ R(t) $ (gallons per hour) | 8.2 | 7.8 | 8.6 | 9.3 |

A. 220.8
B. 19.2
C. 100.8
D. 18.75

The trapezoidal approximation involves dividing the time interval into subintervals and approximating the change in volume as the sum of trapezoidal areas.

Let's calculate the approximate volume of oil at t = 14 hours using the given data and the trapezoidal approximation: Interval 1 (2 to 5 hours):

Average rate = (R(2) + R(5)) / 2 = (8.2 + 7.8) / 2 = 16 / 2 = 8 gallons per hour.

Volume change =

[tex]Average rate \times time = 8 \times (5 - 2)[/tex]

= 24 gallons.

Interval 2 (5 to 10 hours):

Average rate = (R(5) + R(10)) / 2 = (7.8 + 8.6) / 2 = 16.4 / 2 = 8.2 gallons per hour

Volume change =

[tex]Average rate \times time = 8.2 \times (10 - 5) [/tex]

= 41 gallons

Interval 3 (10 to 14 hours):

Average rate = (R(10) + R(14)) / 2 = (8.6 + 9.3) / 2 = 17.9 / 2 = 8.95 gallons per hour

Volume change =

[tex]Average rate \times time = 8.95 \times (14 - 10)[/tex]

= 35.8 gallons.

Total volume change = Interval 1 + Interval 2 + Interval 3 = 24 + 41 + 35.8 = 100.8 gallons.

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