Related rates of change Problem set B

If \(y=x^3-4x^2\), find \(\frac{dy}{dt}\) when \(x=3\) and \(\frac{dx}{dt}=-2\).

If \(y=e^{-2z}\), find the rate of increase of y when \(z=1\) and \(\frac{dz}{dt}=5\).

For each part, find the rate of increase of a.

1. (a) \(a=\frac{1}{b}\), when \(b=5\) and b is increasing at 4 per second.
2. (b) \(a=\frac{1}{b^2}\), when \(b=3\) and b is increasing at 6 per second.

If \(y=\ln(x+4)\), find the rate of increase of y when \(x=2\) and x is decreasing at 3 units per hour.

Given that \(B=x^3+y^3\), find \(\frac{dB}{dt}\) when \(x=1\), \(y=3\), \(\frac{dx}{dt}=-2\) and \(\frac{dy}{dt}=1\).

Given that \(C=\frac{x}{y}\), find \(\frac{dC}{dt}\) when \(x=8\), \(y=4\), \(\frac{dx}{dt}=2\) and \(\frac{dy}{dt}=-1\).

A rectangular garden bed is being reshaped. At one moment its length is 6 m and its width is 4 m. The length is decreasing at 0.2 m per day and the width is increasing at 0.3 m per day. Find the rate of change of the area at this moment, and say whether the area is increasing or decreasing.

The light intensity I units at a depth of d metres in a lake is modelled by \(I=80e^{-0.4d}\). A diver descends at 0.5 m s\(^{-1}\). Find the rate of change of the intensity when \(d=3\).

The sound intensity of a speaker at a distance d metres is \(I=\frac{45}{d^2}\) units. A cyclist rides away from the speaker at 1.5 m s\(^{-1}\). Find the rate of change of the intensity when \(d=3\).

Two square rugs have sides x m and y m. Their total area \(A=x^2+y^2\) stays constant while x is increasing at 3 m per day. Find the rate of change of y when \(x=8\) and \(y=6\).

A photograph is enlarged on a printer. It is x cm wide and y cm high, and the ratio \(C=\frac{x}{y}\) stays constant. When \(x=4\) and \(y=3\), the width is increasing at 1.5 cm s\(^{-1}\). Find the rate of change of the height.