# Math- Calculus

posted by Olivis

A 20 foot ladder is sliding down a vertical wall at a constant rate of 2 feet per second.

a) How fast is the ladder moving away from the wall when the base of the ladder is 12 feet away from the wall?

b) Find the rate of change at which the angle between the ladder and the ground is changing when the base of the ladder is 16 feet from the wall.

1. Reiny

Did you make your sketch?
label:
foot of ladder is x ft from wall
top of ladder is y ft above ground
length of ladder (hypotenuse) is 20

x^2 + y^2 = 400
2x dx/dt + 2y dy/dt = 0

when x = 12
144 + y^2= 400
y = 16

24 dx/dt + 32(-2) = 0
dx/dt = 64/24 or 8/3 ft/s

b) let the angle be Ø
cosØ = x/20
x = 20cosØ
dx/dt = -20sinØ dØ/dt

when x= 16, cosØ = 16/20 = 4/5
then sinØ = 3/5 , (recognize the 3-4-5 right-angled triangle)

using a) when x= 16, y = 12 , dy/dt = -2
32 dxdt + 24(-2) = 0
dx/dt = 48/32 = 3/2

so in dx/dt = -20sinØ dØ/dt
3/2 = -20(3/5) dØ/dt
dØ/dt = (3/2)(5/3)(-1/20) = -1/8

so at that moment, the angle is decreasing at a rate of 1/8 radians/second

2. Reiny

Just noticed that Steve had already answered the same question above.
At least we both ended up with the same answer.

## Similar Questions

1. ### MAth

A 13 foot ladder is resting against the wall. The bottom is initially 10 feet away from the wall and is being pushed towards the wall at a rate of 0.25 ft/sec. How fast is the top of the ladder moving up the wall 20 seconds after we …
2. ### Math- Calculus

A 20 foot ladder is sliding down a vertical wall at a constant rate of 2 feet per second. a) How fast is the ladder moving away from the wall when the base of the ladder is 12 feet away from the wall?
3. ### Math- Calculus

A 20 foot ladder is sliding down a vertical wall at a constant rate of 2 feet per second. a) How fast is the ladder moving away from the wall when the base of the ladder is 12 feet away from the wall?
4. ### Math- Calculus

A 20 foot ladder is sliding down a vertical wall at a constant rate of 2 feet per second. a) How fast is the ladder moving away from the wall when the base of the ladder is 12 feet away from the wall?
5. ### Calculus

a 5m ladder rests against a vertical wall. The top of the ladder begins sliding down the wall at 2m/sec, while the foot of the ladder moves away from the base of the wall. How fast is the foot of the ladder moving away from the wall …
6. ### Calculus

1a. 15 ft ladder is placed against a vertical wall. the bottom of the ladder is sliding away from the wall at a rate of 2 ft/sec. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 9 ft from the …
7. ### Calculus

The top of a 24 foot ladder, leaning against a vertical wall, is slipping down the wall at the rate of 4 feet per second. How fast is the bottom of the ladder sliding along the ground when the bottom of the ladder is 7 feet away from …
8. ### Calculas

A 13ft ladder is placed against a verticle wall. Suppose the bottom of the ladder slides away from the wall at a constant rate of 2 feet per second. How fast is the top of the ladder sliding down the wall (negitive rate) when the bottom …
9. ### Calculus

A ladder 15 meters long leans against a vertical wall. If the foot of the ladder is being pulled away from the wall at the rate of 2 meters per minute. How fast is the top of the ladder sliding down the wall at the instant when the …
10. ### Calculus (related rates)

A ladder 13 meters long rests on horizontal ground and leans against a vertical wall. The foot of the ladder is pulled away from the wall at the rate of 0.4 m/sec. How fast is the top sliding down the wall when the foot of the ladder …

More Similar Questions