# Calculus

posted by .

A 23 foot ladder is leaning on a 6 foot fence. The base of the ladder is being pulled away from the fence at the rate of 7 feet/minute. How fast is the top of the ladder approaching the ground when the base is 6 from the fence? [Note: The ladder is protruding over the top of the fence.]

The top of the ladder is approaching the ground at the rate of____________ feet/minute. [Enter a positive number only.]

• Calculus -

Let the ladder make an angle t with the ground.

Let the distance from the base of the fence = x

Let the end of the ladder be at height h.

tan(t) = 6/x
sin(t) = h/23

tan^2 = sin^2/cos^2

36/x^2 = (h/23)^2 / (1 - (h/23)^2)
flip it upside down:

x^2/36 = (1-h^2/529)/(h^2/529)
x^2/36 = 529/h^2 - 1

2x/36 dx = -2(529)/h^3 dh

when x = 6
tan(t) = 6/6 = 1
so, h = 23/√2

2(6)/36 * 7 = -2(529)/(23^3/2√2) dh

84/36 = -1058/4302 dh

dh = -9.49 ft/s
or,

9.49 ft/s downward

## Similar Questions

1. ### Calculus

related rates: a ladder, 12 feet long, is leaning against a wall. if the foot of the ladder slides away from the wall along level ground,what is the rate of change of the top of the ladder with respect to the distance of the foot of …
2. ### Calculus

A 25 foot ladder is leaning on a 9 foot fence. The base of the ladder is being pulled away from the fence at the rate of 10 feet/minute. How fast is the top of the ladder approaching the ground when the base is 9 from the fence?
3. ### Calculus Ab

A 26 foot step ladder is leaned up against a 30 foot building to reach a window near the top. The ladder is being pushed toward the building in a horizontal direction of 2 feet per second. 1. Find each related rate when the ladder …
4. ### Calculus

A 17 foot ladder is leaning on a 6 foot fence. The base of the ladder is being pulled away from the fence at the rate of 5 feet/minute. How fast is the top of the ladder approaching the ground when the base is 6 from the fence?
5. ### Math Calculus

The foot of a 30-foot ladder is being pulled away from a vertical wall at a rate of 1 foot per minute. When the top of the ladder is 6 feet from the ground, at what rate is the top of the ladder moving down the wall?
6. ### Calculus

Mr. Cook was painting Arvada West High School, standing on top of a 25]foot ladder. Why?
7. ### Calculus

Mr. Cook was painting Arvada West High School, standing on top of a 25]foot ladder. Why?