1. The radius of a sphere increases at a rate of 2 m/sec. Find the rate at which the volume increases when the radius is 10 m.

Ans :\(\frac{dr}{dt}\)=2 m/s              V =\(\frac{4\pi r^2}{3}\)                         \(\frac{dv}{dt}\)=\(4\pi r^2\)\(\frac{dr}{dt}\)= 2513.3 m³/s

2. The radius of a sphere decreases at a rate of 5m/sec. Find the rate at which the surface area decreases when the radius is 20 m.

Ans :\(\frac{dr}{dt}\)=-5 m/s              A =\(4\pi r^2\)                         \(\frac{dA}{dt}\)=\(8\pi r^2\)\(\frac{dr}{dt}\)= -2513.3 m³/s

3. A triangle has a height that is increasing at a rate of 1 m/s and its area is increasing at a rate of 2 m2/s. Find the rate at which the base of the triangle is changing when the height of the triangle is 3 m and the area is 10 m2.

Ans : \(\frac{dr}{dt}\)= 1m/s          \(\frac{dA}{dt}\)= 2m/s  A = \(\frac{1}{2}\)bh  \(\frac{dA}{dt}\) = \(\frac{1}{2}\) (h\(\frac{db}{dt}\) + b\(\frac{dh}{dt}\))

A = 10  h = 3   b = \(\frac{20}{3}\)      \(\frac{db}{dt}\) = –\(\frac{14}{3}\)m/s

4. A vertical cylinder is leaking water at a rate of 2m³/s. If the cylinder has a height of 20 m and a radius of 1 m, at what rate is the height of the water changing when the height is 5 m?

Ans : \(\frac{dv}{dt}\) = 2m³/s     V = \(\pi r^2 h\)    \(\frac{dv}{dt}\)= \(\pi r^2\)\(\frac{dh}{dt}\)                \(\frac{dh}{dt}\)  =- \(\frac{2}{\pi}\) m/s

5.You are stationary on the ground and are watching a bird fly horizontally at a rate of 15 m/s. The bird is located 20 m above your head. How fast does the angle of elevation change when the horizontal distance between you and the bird is 10 m?

Ans :    \(\frac{dx}{dt}\) = 15 m/s   tanΘ = \(\frac{y}{x}\)     y = 20  \(\frac{dΘ}{dt}\) sec²Θ = – \(\frac{y}{x_2}\)\(\frac{dx}{dt}\)

\(\frac{dΘ}{dt}\) = –\(\frac{20}{100}\) x \(\frac{1}{5}\) x 15  = –\(\frac{3}{5}\) rd/s