#### To determine

**To find:** The length of the ladder.

#### Answer

The length of the ladder is L=5 m.

#### Explanation

**Given:**

The top of the ladder slides down a vertical wall at a rate of 0.15 m/s and at the same moment the bottom of the ladder slides away from the wall at a rate of 0.2 m/s.

**Formula used:**

(1) Chain rule: dydx=dydu⋅dudx

(2) Pythagorean Theorem.

**Calculation:**

Let us assume that L be the length of the ladder and x be the horizontal distance from the wall to the lower end of the ladder, and y be the vertical distance from the ground to the top end of the ladder as shown in the Figure 1 given below.

Since the ladder slides away from the wall so the distance x become increases and y decreases.

Therefore x and y are function of the time t

And given that dxdt=0.2 m/s and dydt=−0.15 m/s.

Obtain L when x=3 m from the wall.

By using Pythagorean Theorem in △ABC

x2+y2=L2

Differentiate x2+y2=L2 with respect to the time t

ddt[x2+y2]=ddt[L2]2xdxdt+2ydydt=0 [Qdydx=dydu⋅dudx]2xdxdt=−2ydydtxdxdt=−ydydt

Substitutes dxdt=0.2 and dydt=−0.15 in above.

x×0.2=−y×(−0.15)0.2x=0.15yy=0.20.15x

Substitutes x=3 in y.

y=0.20.15×3y=0.60.15=4 m

Now substitutes x=3 and y=4 in x2+y2=L2

L=32+42=9+16=25=5 m

Therefore, the length of the ladder is L=5 m.