#### To determine

**(a)**

**To find:**

Work done in pulling the rope to the top of the building

#### Answer

W=625 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 rope 50 ft

ii) Height of the building 120 ft

**3) Definition 4:**

W=limn→∞∑t=1nfxi* ∆x=∫abfx dx

**4) Calculation:**

Let’s place the origin at the top of the building and x-axis is pointing downward as in the figure.

Divide the rope into small parts with length ∆x

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

The rope weighs 0.5 pounds per foot, therefore,

The weight of the ith part is (0.5 lb/ft)(∆x ft)=0.5∆x lb=12∆x lb

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

12∆x·xi*=12xi* ∆x

( as 0.5∆x is force and xi* is distance)

Total work done by adding all these approximations and letting the number of parts become large (∆x→0)

W=limn→∞∑i=1n12xi*∆x=∫05012xdx

=14x2050

=25004

=625 ft·lb

**Conclusion:**

Work done in pulling the rope to the top of building is 625 ft·lb

#### To determine

**(b)**

**To find:**

Work done in pulling half the rope to the top of the building

#### Answer

18754 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 rope 50 ft

ii) Weights of rope 0.5 lb/ft

**3) Calculation:**

Half the rope is 502=25 ft

When half the rope is pulled to the top of the building, the work to lift the top half of the rope is W1=∫02512xdx=14x2025=6254 ft

The bottom half of the rope is lifted 25 ft and the work needed to accomplish that is

W2=∫255012·25dx=252x2550=6252 ft

The work done in pulling half the rope to the top of the building is

W=W1+W2=6254+6252=18754 ft

**Conclusion:**

Work done in pulling half the rope to the top of building is 18754 ft·lb