Related rates of change Problem set A

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

If \(y=e^{4z}\), find the rate of increase of y when \(z=0.5\) and \(\frac{dz}{dt}=3\).

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

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

If \(y=\ln(x+2)\), find the rate of increase of y when \(x=4\) and x is decreasing at 5 units per hour.

Given that \(B=x^3+y^3\), find \(\frac{dB}{dt}\) when \(x=2\), \(y=1\), \(\frac{dx}{dt}=3\) and \(\frac{dy}{dt}=-4\).

Given that \(C=\frac{x}{y}\), find \(\frac{dC}{dt}\) when \(x=6\), \(y=2\), \(\frac{dx}{dt}=-1\) and \(\frac{dy}{dt}=3\).

A sheet of dough is rolled out as a rectangle. At one moment its length is 30 cm and its width is 20 cm. The length is increasing at 3 cm s\(^{-1}\) and the width is decreasing at 0.5 cm s\(^{-1}\). Find the rate of change of the area at this moment, and say whether the area is increasing or decreasing.

Atmospheric pressure P kPa at a height of h km is modelled by \(P=101e^{-0.12h}\). A weather balloon rises at 0.3 km per minute. Find the rate of change of the pressure when \(h=5\).

The brightness of a lamp at a distance d metres is \(I=\frac{80}{d^2}\) lux. A student walks away from the lamp at 0.5 m s\(^{-1}\). Find the rate of change of the brightness when \(d=4\).

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

An image on a screen is x cm wide and y cm high, and the ratio \(C=\frac{x}{y}\) stays constant as it is resized. When \(x=16\) and \(y=9\), the width is increasing at 2 cm s\(^{-1}\). Find the rate of change of the height.