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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Oil leaked from a tank at a rate of r(t) liters per hour. The rate decreased as time passed and values of the rate at two-hour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out.

Chapter 5.1, Problem 15E, Oil leaked from a tank at a rate of r(t) liters per hour. The rate decreased as time passed and

To determine

The lower estimate and upper estimate for the total amount of oil that leaked out.

Explanation

Given:

The function r(t) is an decreasing function.

The number of intervals is n=5.

Each interval value is Δt=2h.

Calculation:

The distance approximation R5 provides the lower estimate for the total amount of oil leaked from the tank and L5 provides the upper estimate for the total amount of oil leaked from the tank.

The expression to find the lower estimate for the total amount of oil leaked (Rn) is shown below:

Rn=r(t1)Δt+r(t2)Δt+...+r(tn)Δt (1)

Here, for the time t1, the rate is r(t1), the time interval is Δt, for the time t2, the rate is r(t2), and for the time tn, the rate is r(tn).

Substitute 5 for n in Equation (1).

R5=r(t1)Δt+r(t2)Δt+r(t3)Δt+r(t4)Δt+r(t5)Δt (2)

Substitute 7.6L/h for r(t1), 2h for Δt, 6.8L/h for r(t2), 6.2L/h for r(t3), 5.7L/h for r(t4), and 5.3L/h for r(t5) in Equation (2).

R5=(7.6×2)+(6.8×2)+(6.2×2)+(5.7×2)+(5.3×2)=15

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