Rates of Change
Metal Cube
- \(V=x^3\)
- \(\frac{dV}{dx}=3x^2\)
- \(\frac{dx}{dt}=\frac{dx}{dV}\times \frac{dV}{dt}\)
- \(\frac{dx}{dV}=\frac{1}{3x^2}\)
- \(\frac{dx}{dt}=\frac{1}{3\times8^2}\times0.048=2.5\times10^{-4}\)
- \(\frac{dA}{dx}6x^2=12x\)
- \(\frac{dA}{dx}=96\)
- \(\frac{dA}{dt}=\frac{dA}{dx}\times \frac{dx}{dt}\)
- \(96\times2.5\times10^{-4}=0.024cm^2s^{-1}\)
Q1
- \(\frac{dA}{dt}=1.5\)
- \(\frac{dr}{dt}=\frac{dr}{dA}\times \frac{dA}{dt}\)
- \(\frac{1}{2\pi \sqrt{ \frac{2}{\pi} }}\times1.5=0.299\)
Q2
- \(\frac{dv}{dt}=3\)
- \(\frac{dr}{dt}=\frac{dr}{dV}\times \frac{dV}{dt}\)
- \(V=\frac{4}{3}\pi r^3\)
- \(\frac{dV}{dr}=4\pi r^2\)
- \(\frac{dr}{dt}=\frac{1}{4\pi r^2}\)
- \(\frac{dr}{dt}=\frac{1}{4\pi 4^2}\times3=0.0149\)
- \(\frac{dS}{dt}=\frac{dS}{dr}\times \frac{dr}{dt}\)
- \(\frac{dS}{dt}=8\pi \times4\times0.0149=1.50\)