Q1. A ladder 15 m long rests against a vertical wall. The top slides down the wall whilst the bottom moves away at the rate of 2m/s. How fast is the angle between the top of the ladder and the wall is changing when the angle is π/3 rd?

Ans : \(\frac{dx}{dt}\) = 2 m/s        SinΘ = \(\frac{x}{15}\)     cosΘ \(\frac{dΘ}{dt}\) = \(\frac{x}{15}\)    \(\frac{dΘ}{dt}\) = \(\frac{4}{15}\) rd/s

Q2. A plane is flying horizontally at an altitude of 3 km and at a speed of 480 km/h passes directly over an observer on the ground. How fast is the distance from the observer to the plane changing 30 s later?

Ans : \(\frac{dx}{dt}\) = \(\frac{400}{3}\) m/s       x²+ 3000² = z²              2x\(\frac{dx}{dt}\) = 2 z \(\frac{dz}{dt}\)            \(\frac{dz}{dt}\) = \(\frac{300}{3}\)m/s

Q3. Consider a cube of variable size. Its length is increasing. If the volume is increasing at the rate of 20 m³/mn. How fast is the surface area increasing when its side length is 8 m?

Ans :       v = a³         s = 6 a²          \(\frac{dv}{dt}\) = 3a² \(\frac{da}{dt}\)    \(\frac{ds}{dt}\)  = 12a\(\frac{da}{dt}\)

\(\frac{a}{t}\) = \(\frac{4}{a}\)\(\frac{dv}{dt}\) = \(\frac{4}{8}\)20 = 10m³/mn

Q4. A girl starts walking north at a speed of 1.5 m/s and a boy starts west from the same point at 2 m/s. At what rate is the distance between the boy and the girl increasing 6 s later?

Ans : x² + y²= a²     \(\frac{dy}{dt}\)y + x\(\frac{dx}{dt}\) + a\(\frac{da}{dt}\)        \(\frac{da}{dt}\) = \(\frac{1}{15}\) (1.5 x 9 + 2* 12) = 2.5 m/s

Q5. Sand is pouring out of a tube at 1 m³/s. It forms a cone-shaped pile. The height of the cone is equal to the radius of its base. How fast is the sandpile rising when its 2 m high?

Ans: v = \(\pi r^2h\) =  \({\pi}h^3\)      \(\frac{dv}{dt}\) = \(3{\pi}h^2\)\(\frac{dh}{dt}\)                  \(\frac{dh}{dt}\) = \(\frac{1}{3{\pi}4}\) 0.027 m/s