#### To determine

**(a)**

**To find:**

Work done if the winch winds up 25 ft of the cable

#### Answer

W=3562.5 ft-lb

#### Explanation

**1) Concept:**

Approximate the required work by using the concept of Riemann sum. Then express the work as an integral and evaluate it.

**2) Given:**

i) Length of cable 60 ft

ii) Weights of cable 180 lb

**3) Definition 4:**

W=limn→∞∑t=1nfxi* ∆x=∫abfx dx

**4) Calculation:**

Use an argument similar to the one that led to Definition 4

Let’s place the origin at the top of the crane and x-axis pointing downward as in the figure. Divide the cable into small parts with length ∆x

If xi* is a point in the ith such interval, then all the points in the interval are lifted by approximately the same amount,xi*.

Total length of the cable 60 ft

Weights of cable 180 lb

Therefore, the cable weights 180/60=3 pounds per foot

If xi*<25ft, then the i th part has to be lifted roughly xi* ft. If xi*≥25 ft, then the i th part has to be lifted 25ft.

The weight of the ith part is 3lb/ft∆x ft=3 ∆x lb

Thus, the work done on the ith part, in foot-pounds, is

3 ∆x·xi*=3xi* ∆x, if xi*<25ft and 3 ∆x25=75∆x if xi*≥25ft

The work of lifting the top 25 ft of the cable is

W1=limn→∞∑i=1n13xi* ∆x=∫0253xdx=32x2025=18752=937.5 ft-lb

n1 represents the number of parts of the cable in the top 25 ft.

The work of lifting the bottom 35 ft of the cable is

W2=limn→∞∑i=1n275 ∆x=∫256075dx=75 x2560=7560-25=2625 ft-lb

Where n2 represents the number of small parts in the bottom 35 feet of the cable

Total work done is

W=W1+W2=937.5+2625=3562.5 ft-lb

**Conclusion:**

Work done if the winch winds up 25 ft of the cable is 3562.5 ft-lb

#### To determine

**(b)**

**To estimate:**

Work done when the winch pulls up ∆x ft of the cable

#### Answer

W=3562.5 ft-lb

#### Explanation

**1) Concept:**

Approximate the required work by using the concept of Riemann sum. Then express the work as an integral and evaluate it.

**2) Given:**

i) Length of the cable 60 ft

ii) Weight of the cable 180 lb

**3) Calculation:**

Divide the cable into small parts with length ∆x

Total length of the cable 60 ft

Weight of the cable 180 lb

Therefore, the cable weighs 180/60=3 pounds per foot

Let x feet of cable be wound up by the winch,

Remaining 60-xft of the cable is still hanging from the winch

Therefore, the portion 60-xft of cable weighs 360-x lb.

Work done on the 60-xft of cable, in foot-pounds, is 360-x ∆x ft-lb

Thus, the total work needed to lift the 25 ft cable is

W=∫025360-x dx=180x-32x2025=18025-32252=4500-937.5=3562.5 ft-lb

**Conclusion:**

The work done when the winch pulls up ∆x *ft* of cable is 3562.5 ft-lb