Chapter 6.1, Problem 28E

(a)

To determine

The upper and lower estimate of the total quantity of the oil that has escaped after 5 hours.

Answer to Problem 28E

The upper estimate is 758 gal and the lower estimate is 543 gal.

Explanation of Solution

Given information:

  

Formula used:

Add the leakages and multiply with unity because length of subinterval is unity.

Calculation:

For upper estimate after 5 hour take the value of leakage from time equal to 1 up to 5, Then Add the leakages and multiply with unity because length of subinterval is unity.

  $\begin{array}{l}\text{upper estimate}=1\left[70+97+136+190+265\right]\\ \text{upper estimate}=758\text{gal}\end{array}$

For lower estimate after 5 hour take the value of leakage from time equal to 0 up to 4, Then Add the leakages and multiply with unity because length of subinterval is unity.

  $\begin{array}{l}\text{lower estimate}=1\left[50+70+97+136+190\right]\\ \text{lower estimate}=543\text{gal}\end{array}$

Conclusion:

The upper estimate is 758 gal and the lower estimate is 543 gal.

(b)

To determine

The quantity of the oil that has escaped after 8 hrs.

Answer to Problem 28E

The upper estimate is 2363 gal and the lower estimate is 1693 gal.

Explanation of Solution

Given information:

  

Formula used:

Add the leakages and multiply with unity because length of subinterval is unity.

Calculation:

For upper estimate after 8 hour take the value of leakage from time equal to 1 up to 8, Then Add the leakages and multiply with unity because length of subinterval is unity.

  $\begin{array}{l}\text{upper estimate}=1\left[70+97+136+190+265+369+516+720\right]\\ \text{upper estimate}=2363\text{gal}\end{array}$

For lower estimate after 5 hour take the value of leakage from time equal to 0 up to 4, Then Add the leakages and multiply with unity because length of subinterval is unity.

  $\begin{array}{l}\text{lower estimate}=1\left[50+70+97+136+190+265+369+516\right]\\ \text{lower estimate}=1693\text{gal}\end{array}$

Conclusion:

The upper estimate is 2363 gal and the lower estimate is 1693 gal.

(c)

To determine

The number of hours that will elapse in the worst case before the oil has been leaked.

Answer to Problem 28E

The time taken will be 32.37 hours.

Explanation of Solution

Given information:

  

Formula used:

The time taken is determined by dividing the leakage with time.

Calculation:

In the worst case,

  $\begin{array}{l}\text{rest}\text{of}\text{oil}=25000-2363\\ \text{rest}\text{of}\text{oil}=22,637\text{gal}\\ \text{time taken}=\frac{22637}{720}\\ \text{time taken}=31.44\text{hours}\end{array}$

In the best case,

  $\begin{array}{l}\text{rest}\text{of}\text{oil}=25000-1693\\ \text{rest}\text{of}\text{oil}=22,307\text{gal}\\ \text{time taken}=\frac{22307}{720}\\ \text{time taken}=32.37\text{hours}\end{array}$

Conclusion:

The time taken will be 32.37 hours.