Answer to Question #305297 in Calculus for Anniee

Question

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?

Answer 1

1.

$$x^2+y^2=(20)^2$$

Differentiate both sides with respect to $t$

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

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

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

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

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

$$dx/dt=\sqrt{3}\ ft/s$$

2.

$$V=\dfrac{4\pi}{3} r^3$$

Differentiate with respect to $t$

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

$$dr/dt=\dfrac{1}{4\pi r^2}dV/dt$$

$$dr/dt=\dfrac{1}{4\pi (7m)^2}(0.2m^3/s)\approx0.000325\ m/s$$