High School

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

Answer :

The estimate of the number of gallons of oil in the tank at t = 14 hours is 100.8 gallons. The correct answer is option C.

To estimate the number of gallons of oil in the tank at t = 14 hours using a trapezoidal approximation,

we need to calculate the total change in oil volume over the given time period.

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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