# Calculus

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.]

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1. 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

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