(a) Given that \(\log_{10} p = a, \log_{10} q = b\) and \(\log_{10} s = c\), express \(\log_{10} (\frac{p^{\frac{1}{3}}q^{4}}{s^{2}}\) in terms of a, b and ...
Question 1 Report
(a) Given that \(\log_{10} p = a, \log_{10} q = b\) and \(\log_{10} s = c\), express \(\log_{10} (\frac{p^{\frac{1}{3}}q^{4}}{s^{2}}\) in terms of a, b and c.
(b) The radius of a circle is 6cm. If the area is increasing at the rate of 20\(cm^{2}s^{-1}\), find, leaving the answer in terms of \(\pi\), the rate at which the radius is increasing.
Answer Details
(a) Using the laws of logarithms with \(\log_{10}p=a,\ \log_{10}q=b,\ \log_{10}s=c\):
\[\log_{10}\!\left(\frac{p^{\frac{1}{3}}q^{4}}{s^{2}}\right)=\tfrac{1}{3}\log_{10}p+4\log_{10}q-2\log_{10}s=\frac{a}{3}+4b-2c\]
(b) Area of a circle: \(A=\pi r^{2}\). Differentiate with respect to time:
\[\frac{dA}{dt}=2\pi r\,\frac{dr}{dt}\]
With \(\dfrac{dA}{dt}=20\,\text{cm}^2\text{s}^{-1}\) and \(r=6\,\text{cm}\):
\[20=2\pi(6)\frac{dr}{dt}\ \Rightarrow\ \frac{dr}{dt}=\frac{20}{12\pi}=\frac{5}{3\pi}\ \text{cm s}^{-1}\]
Download The App On Google Playstore
Everything you need to excel in your exams
