A circle has radius \(r\) cm increasing at \(\frac{dr}{dt}=2\) cm/s. Find \(\frac{dA}{dt}\) when \(r=6\) cm, where \(A=\pi r^2\).

A cube has side length \(s\) cm increasing at \(\frac{ds}{dt}=0.5\) cm/s. Find \(\frac{dV}{dt}\) when \(s=4\) cm, where \(V=s^3\).

For a right triangle with legs \(x\) cm and \(y\) cm, \(x^2+y^2=25\). If \(\frac{dx}{dt}=1\) cm/s, \(x=3\) cm, and \(y=4\) cm, find \(\frac{dy}{dt}\).

A spherical balloon has radius \(r\) cm increasing at \(3\) cm/s. Find \(\frac{dV}{dt}\) when \(r=10\) cm, where \(V=\frac{4}{3}\pi r^3\).

A ladder \(13\) m long leans against a wall. The foot of the ladder moves away from the wall at \(2\) m/s. Find how fast the top moves down when the foot is \(5\) m from the wall.

The area of a rectangle is \(A=xy\), where \(x\) and \(y\) are side lengths in cm. At one instant \(x=4\), \(y=7\), \(\frac{dx}{dt}=3\) cm/s, and \(\frac{dy}{dt}=-1\) cm/s. Find \(\frac{dA}{dt}\).

Water flows into a cone-shaped tank with volume \(V=\frac{1}{3}\pi r^2h\). The tank shape gives \(r=\frac{h}{2}\). If \(\frac{dV}{dt}=6\pi\) cm\(^3\)/s, find \(\frac{dh}{dt}\) when \(h=4\) cm.

A point moves along the curve \(y=x^2+1\). If \(\frac{dx}{dt}=4\) units/s, find \(\frac{dy}{dt}\) when \(x=3\).

Two cars leave the same point at the same time. One travels east at \(60\) km/h and the other north at \(80\) km/h. Find the rate at which the distance between the cars is increasing after \(1\) hour.

For a circle with radius \(r\) cm, the circumference is \(C=2\pi r\). If \(\frac{dC}{dt}=5\pi\) cm/s, find \(\frac{dr}{dt}\).