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 North 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.
(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.
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Correct Answer: Option n
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 = \(\frac{1200}{0.25} = N4800\)
\(0.8x = N4,800\)
\(x = \frac{4800}{0.8} = N6000\)
(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 = \(\frac{1200}{0.25} = N4800\)
\(0.8x = N4,800\)
\(x = \frac{4800}{0.8} = N6000\)