#### To determine

**To find:** The rate at which the cart B is moving toward the point Q when cart A is 5 ft from Q.

#### Answer

The rate at which the cart B is moving toward the point Q is dydt=−10133≈−0.87 ft/s

#### Explanation

**Given:**

The two carts connected by a rope 39 ft long and pass over the pulley P .

The point Q is on the floor 12 ft directly below the point P and between the carts.

The rate of moving of the cart A away from the point Q is 2 ft/s.

**Formula used:**

(1) Chain rule:

(2) Pythagorean Theorem.

**Calculation:**

The two carts A and B connected by the 39 ft long rope and pass over the pulley P.

Let us assume that x be the distance between the cart A and the point Q and y be the distance between the cart B and the point Q as shown the figure 1 given below.

Let consider Q as the origin point.

Since the distance x and y changes with the time t.

Therefore, x and y are the function of the time t.

Since dxdt=−2 ft/s.

Obtain dydt when x=5 ft from Q.

From the above figure 1 the total length is.

AB=AP+PB .

Then by using the Pythagorean Theorem in the triangle △AQP and △BQP

AB=x2+122+y2+122 .

Since the total length of the rope AB= 39 ft, therefore

x2+122+y2+122=39

Differentiate the total length of the rope with respect to the time t.

ddt[x2+122+y2+122]=ddt[39]ddx[x2+122]dxdt+ddy[y2+122]dydt=0 [Qdydx=dydu⋅dudx][12(x2+122)12 −1(2x)]dxdt+[12(y2+122)12 −1(2y)]dydt=0[xx2+122]dxdt+[yy2+122]dydt=0

On further simplification the rate of change speed of the cart Q is as follows.

[yy2+122]dydt=−[xx2+122]dxdtdydt=−xyy2+122x2+122dxdt

Substitutes x=−5 in total length and find y as follows.

(−5)2+122+y2+122=3925+144+y2+122=39169y2+122=3913+y2+122=39

On further simplification the value of y is as follows.

y2+122=39−13y2+122=26y2+122=262y2=262−122

On simplification the value of y is as follows.

y2=676−144y2=532y=532 ft

Substitutes x=−5,y=532 and dxdt=−2 in dydt .

dydt=−(−5)532532+12225+122(−2)=−10532676169=−10532(2613)=−10532(2) ft/s

On further simplification the rate of change of the speed of cart Q is as follows.

dydt=−10532(2)=−102133(2)=−10133≈−0.87 ft/s

Therefore, the speed at which the cart b is moving towards Q is dydt=−10133≈−0.87 ft/s.