The top of a ladder slides down a vertical wall at a rate of $0.1$ m/s . At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s . How long is the ladder?
A) $7 \mathrm{~m}$
B) $2.3 \mathrm{~m}$
C) $2 \mathrm{~m}$
D) $2.8 \mathrm{~m}$
E) None of these

Question

Quizplus · Accepted Answer

The ladder is forming a right triangle with the wall and the ground. The rate of change of the area of the triangle is the product of the rates of change of the two sides that form the right angle. So, the rate of change of the area is $0.1 \times 0.2 = 0.02$ m^2/s. The area of the triangle is $\frac{1}{2} \times \text{base} \times \text{height}$. So, the rate of change of the area is also $\frac{1}{2} \times \text{rate of change of base} \times \text{height} + \frac{1}{2} \times \text{base} \times \text{rate of change of height}$. Setting these two expressions equal to each other, we get $0.02 = \frac{1}{2} \times 0.2 \times \text{height} + \frac{1}{2} \times 3 \times 0.1$. Solving for the height, we get $\text{height} = 7 \mathrm{~m}$. So, the length of the ladder is $7 \mathrm{~m}$.

The Top of a Ladder Slides Down a Vertical Wall 0.10.10.1

The Top of a Ladder Slides Down a Vertical Wall $0.1$