#### To determine

**To find:** The speed at which the boat is approaching the dock when it is 8 m from the dock.

#### Answer

The boat approaches the dock at 658≈1.01 m/s.

#### Explanation

**Given:**

Since the rope is pulled at a rate of 1 m/s, that is dydt=−1 m/s.

**Formula used:**

Chain rule: dydx=dydu⋅dudx

**Calculation:**

Let x be the distance between the boat and dock and y be the length of the rope which is attached to the bow of the boat and passing through a pulley on the dock as shown in the figure-1 given below.

Since x and y decreases with the time t, both x and y are functions of the time t.

Obtain the value of dxdt when the boat is 8 m from the dock.

Apply Pythagorean Theorem in Figure 1, y2=x2+1

If x=8, then the value of *y* is obtained as follows.

y=82+1=64+1=65 m

Differentiate y with respect to the time t.

ddt[y2]=ddt[x2+1]2ydydt=2xdxdt [∵dydx=dydu⋅dudx ]dxdt=yx⋅dydt

Substitute x=8, y=65 and dydx=−1 in dxdt.

dxdt=(658)(1)=658 m/s

Therefore, the boat approaches the dock at 658≈1.01 m/s.