Question #305297

1. A ladder 20 ft long leans against a vertical building. If the top of the ladder slides down at a rate of p 3 ft s, how fast is the bottom of the ladder sliding away from the building when the top of the ladder is 10 ft above the ground?



2. A sphere is growing in such a manner that its volume increases at 0:2 m³ s(cubic meter per second). How fast is its radius increasing when it is 7 m long?

1
Expert's answer
2022-03-05T05:04:42-0500

1.


x2+y2=(20)2x^2+y^2=(20)^2

Differentiate both sides with respect to tt


2x(dx/dt)+2y(dy/dt)=02x(dx/dt)+2y(dy/dt)=0

dx/dt=(yx)dy/dtdx/dt=-(\dfrac{y}{x})dy/dt

dx/dt=(y400y2)dy/dtdx/dt=-(\dfrac{y}{\sqrt{400-y^2}})dy/dt

Given dy/dt=3ft/s,y=10ftdy/dt=-3ft/s, y=10ft


dx/dt=(10400(10)2)(3ft/s)dx/dt=-(\dfrac{10}{\sqrt{400-(10)^2}})(-3ft/s)

dx/dt=3 ft/sdx/dt=\sqrt{3}\ ft/s

2.

V=4π3r3V=\dfrac{4\pi}{3} r^3

Differentiate with respect to tt


dV/dt=4π3(3r2)dr/dtdV/dt=\dfrac{4\pi}{3}(3 r^2)dr/dt

dr/dt=14πr2dV/dtdr/dt=\dfrac{1}{4\pi r^2}dV/dt

dr/dt=14π(7m)2(0.2m3/s)0.000325 m/sdr/dt=\dfrac{1}{4\pi (7m)^2}(0.2m^3/s)\approx0.000325\ m/s



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