How fast is the top moving down the wall when the base of the ladder is 7 feet away from the wall?

Question

Other given information:

A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second.

Answer 1

PROBLEM

How fast is the top moving down the wall when the base of the ladder is 7 feet away from the wall?

Other given information:

A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second.

SOLUTION

Diagram

|

|

| ↓ dy/dt = ?

|\

| \

| \ Ladder, 25 ft = c

| \

| \

|_____\_→ dx/dt = 2 ft/s

↔

7 ft

Using Pythagorean Theorem, the height of the top of ladder when the base is 7 feet from the wall:

x² + y² = c²

7² + y² = 25²

y² = 576

y = √y² = √576

y = 24 ft

Taking derivative of Pythagorean Theorem equation to find rates of change:

(d/dt)[x² + y² = c²]

2x(dx/dt) + 2y(dy/dt) = 2c(dc/dt)

dc/dt = 0 ft/s, length of ladder does not change

dx/dt = 2 ft/s

x = 7 ft

y = 24 ft

c = 25 ft

2(7)(2) + 2(24)(dy/dt) = 2(25)(0)

28 + 48(dy/dt) = 0

48(dy/dt) = -28

dy/dt = -28/48

dy/dt = -7/12 ft/s