Related rates of change Problem set B One question at a time

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

• \(\frac{dy}{dx}=3 x^{2} - 8 x\)
• \(\frac{dy}{dt}=(3 x^{2} - 8 x)\frac{dx}{dt}\)
• \(=(3)(-2)=-6\)

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

• \(\frac{dy}{dz}=-2e^{-2z}\)
• \(\frac{dy}{dt}=-2e^{-2}\times5\)
• \(=-1.35\)

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.

   • \(a=b^{-1}\)
   • \(\frac{da}{db}=-b^{-2}=-\frac{1}{b^2}\)
   • \(\frac{da}{dt}=-\frac{1}{5^2}\times4=-0.16\)

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

   • \(a=b^{-2}\)
   • \(\frac{da}{db}=-2b^{-3}=-\frac{2}{b^3}\)
   • \(\frac{da}{dt}=-\frac{2}{3^3}\times6=-4/9 = -0.444\)

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

• \(\frac{dx}{dt}=-3\)
• \(\frac{dy}{dx}=\frac{1}{x+4}\)
• \(\frac{dy}{dt}=\frac{1}{6}\times(-3)=-0.5\)

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

• \(\frac{dB}{dt}=3x^2\frac{dx}{dt}+3y^2\frac{dy}{dt}\)
• \(=3(1)^2(-2)+3(3)^2(1)\)
• \(=21\)

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

• \(\frac{dC}{dt}=\frac{y\frac{dx}{dt}-x\frac{dy}{dt}}{y^2}\)
• \(=\frac{(4)(2)-(8)(-1)}{4^2}\)
• \(=1\)

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.

• \(A=xy\)
• \(\frac{dA}{dt}=x\frac{dy}{dt}+y\frac{dx}{dt}\)
• \(=(6)(0.3)+(4)(-0.2)=1\)
• The area is increasing at 1 m² per day.

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 \(d=3\).

• \(\frac{dP}{dh}=-32e^{-0.4h}\)
• \(\frac{dP}{dt}=\frac{dP}{dh}\times\frac{dh}{dt}=-32e^{-1.2}\times0.5\)
• \(=-4.82\)

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 \(d=3\).

• \(I=45d^{-2}\)
• \(\frac{dI}{dd}=-90d^{-3}=-\frac{90}{d^3}\)
• \(\frac{dI}{dt}=-\frac{90}{3^3}\times1.5=-5.00\)

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 \(y=6\).

• \(A=x^2+y^2\)
• \(\frac{dA}{dt}=2x\frac{dx}{dt}+2y\frac{dy}{dt}=0\)
• \(2(8)(3)+2(6)\frac{dy}{dt}=0\)
• \(\frac{dy}{dt}=-\frac{48}{12}=-4\)

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.

• \(C=\frac{x}{y}\)
• \(\frac{dC}{dt}=\frac{y\frac{dx}{dt}-x\frac{dy}{dt}}{y^2}=0\)
• \((3)(1.5)-(4)\frac{dy}{dt}=0\)
• \(\frac{dy}{dt}=\frac{4.5}{4}=1.125\)