Please show all steps and give all answers in exact form! Thanks! The height of

Question

Please show all steps and give all answers in exact form! Thanks!

The height of a triangle decreases at a rate of 5 feet per minute while the area of the triangle increases at a rate of 8 square feet per minute. At what rate is the base of the triangle changing when the height is 12 feet and the area is 144 square feet? Do not assume you have a right triangle. A 24 foot ladder is leaning against a wall. The top of the ladder is sliding down the wall at a rate of 1/2 foot per second. How fast is the bottom of the ladder moving along the ground when the bottom of the ladder is 6 feet from the wall? How fast is the angle 0 changing when the bottom of the ladder is 6 feet from the wall? A man 6 feet tall is walking away from a lamppost 20 feet high at a rate of 5 feet per second. The light at the top of the lamppost 20 feet above the ground is casting a shadow of the man. At what rate is the tip of the man's shadow moving when he is 10 feet from the base of the lamppost? At what rate is the length of the man's shadow changing when he is 10

Explanation / Answer

1) Area = 0.5 * b*h

so dA/dt = 0.5 * { bdh/dt + hdb/dt)

given ,

dh/dt = - 5

dA/dt = 8

A = 144

h = 12

so b = 144 /(0.5*12) = 24 ft

so 8 = 0.5 * { 34*(-5) + 12*db/dt }

=> db/dt = 15.5 ft/min

2) x^2 + y^2 = 24^2 = 576

a) differentiating both sides wrt t

2xdx/dt + 2ydy/dt = 0

=> dy/dt = - (x/y) dx/dt

given y = 6

dx/dt = - 1/2

so x = sqrt ( 576 - 36) = 23.24 ft

so dy/dt = - (23.24/6)*(-1/2) = 1.94 ft/s

b) tan(theta) = x/y

sec^2(theta) *d(theta)/dt = (ydx/dt - xdy/dt) /y^2

y = 6 , x = 23.24

dy/dt = 1.94 , dx/dt = -1/2

sec^2(theta) = (24/y)^2 = 16

so 16*d(theta)/dt = (6*-1/2 - 23.24*1.94)/36

=> d(theta)/dt = - 0.083 rad/s

3) from similarity

x/(x+y) = 6/20

=> 1 + y/x = 10/3 => y/x = 7/3

a) speed of man / speed of shadow = height of man /length of shadow

=> 5 /v = 6 / x

given , y = 10

so x = 30/7

so v = 5*30/(6*7) = 3.57 ft/s

b) d(x+y)/dt = 5

=> dx/dt + dy/dt = 5

=> dy/dt = 5 - dx/dt

y/x = 7/3

=> xdy/dt - ydx/dt = 0

=> y = 10

so x = 30/7

=> 30/7 * (5-dx/dt) - 10*dx/dt = 0

=> dx/dt = 1.154 ft/s

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