K(lat. 60°N, long. 50°W) is a point on the eart’s surface. L is another point due East of K and the third point N is due South of K. The distance KL is 3520km and KN is 10951km.
(a) Calculate: (i) The longitude of L ; (ii) The latitude of N. (Take \(\pi = \frac{22}{7}\) and the radius of the earth = 6400km).
(b) A man was allowed 20% of his income as tax free. He then paid 25 kobo in the naira on the remainder. If he paid N1,200.00 as tax, calculate his total income.
Explanation
(a)
(i) Let the length of the longitude be \(\theta\).
Longitude difference = \(50 + \theta\)
Distance along the parallel of latitude (KL) = \(\frac{50 + \theta}{360} \times 2\pi R \cos 60\)
\(3520 = \frac{50 + \theta}{360} \times 2 \times \frac{22}{7} \times 6400 \cos 60\)
\(3520 = \frac{50 + \theta}{360} \times 2 \times \frac{22}{7} \times 3200\)
\(50 + \theta = \frac{3520 \times 360 \times 7}{22 times 2 \times 3200}\)
\(50 + \theta = 63°\)
\(\theta = 63° - 50° = 13°\)
(ii) Let the latitude of N be \(\alpha\).
The distance along the parallel of longitude (KN) = \(\frac{60 + \alpha}{360} \times 2 \times \frac{22}{7} \times 6400\)
\(10951 = \frac{60 + \alpha}{360} \times 2 \times \frac{22}{7} \times 6400\)
\(60 + \alpha = \frac{10951 \times 360 \times 7}{2 \times 22 \times 6400}\)
\(60 + \alpha = 98°\)
\(\alpha = 98° - 60° = 38°\)
The latitude of N = 38°S.
(b) Let the man's income tax be x.
80% of x = \(\frac{1200}{0.25} = N4800\)
\(0.8x = N4,800\)
\(x = \frac{4800}{0.8} = N6000\)