# The lower estimate and upper estimate for the total amount of oil that leaked out. ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 5.1, Problem 13E
To determine

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

Expert Solution

The lower estimate for the total amount of oil leaked is 63.2L.

The upper estimate for the total amount of oil leaked is 70 L.

### Explanation of Solution

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)=15.2+13.6+12.4+11.4+10.6=63.2L.

Therefore, the lower estimate for the total amount of oil leakage from tank is 63.2 L.

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

Ln=r(t0)Δt+r(t1)Δt+...+r(tn1)Δt (3)

Here, for the time t0, the rate is r(t0) and for the time tn1, the rate is r(tn1).

Substitute 5 for n in Equation (3).

L5=r(t0)Δt+r(t1)Δt+r(t2)Δt+r(t3)Δt+r(t4)Δt (4)

Substitute 8.7L/h for r(t0), 2h for Δt, 7.6L/h for r(t1), 6.8 L/h for r(t2), 6.2L/h for r(t3), and 5.7L/h for r(t4) in Equation (4).

L5=(8.7×2)+(7.6×2)+(6.8×2)+(6.2×2)+(5.7×2)=17.4+15.2+13.6+12.4+11.4=70L.

Therefore, the upper estimate for the total amount of oil leakage from tank is 70 L.

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