(a) If \((y - 1)\log_{10}4 = y\log_{10}16\), without using Mathematics tables or calculator, find the value of y. (b) When I walk from my house at 4km/h, I ...
Question 1 Report
(a) If \((y - 1)\log_{10}4 = y\log_{10}16\), without using Mathematics tables or calculator, find the value of y.
(b) When I walk from my house at 4km/h, I will get to my office 30mins later than when I walk at 5km/h. Calculate the distance between my house and office.
Answer Details
(a) Given \((y - 1)\log_{10}4 = y\log_{10}16\).
Since \(16 = 4^2\), we have \(\log_{10}16 = 2\log_{10}4\). Substituting:
\[(y - 1)\log_{10}4 = y(2\log_{10}4).\]
Divide both sides by \(\log_{10}4\) (which is not zero):
\[y - 1 = 2y \Rightarrow -1 = y \Rightarrow y = -1.\]
(b) Let the distance from house to office be \(d\) km. Time at 4 km/h is \(\dfrac{d}{4}\) h; time at 5 km/h is \(\dfrac{d}{5}\) h. Walking at the slower speed takes 30 minutes \(\left(= \tfrac{1}{2}\text{ h}\right)\) longer:
\[\frac{d}{4} - \frac{d}{5} = \frac{1}{2}.\]
\[d\left(\frac{5 - 4}{20}\right) = \frac{1}{2} \Rightarrow \frac{d}{20} = \frac{1}{2} \Rightarrow d = 10.\]
The distance between the house and the office is \(\mathbf{10\text{ km}}\).
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