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 3520...

Answer Details

\(R=6400\) km, \(\pi=\tfrac{22}{7}\). \(K(60^{\circ}\N,50^{\circ}\W)\).

(a)(i) Longitude of L (due east of \(K\), along latitude \(60^{\circ}\N\)). Radius of that parallel uses \(\cos60^{\circ}=\tfrac12\):

\[KL=\frac{\theta}{360}\times2\pi R\cos60^{\circ}\Rightarrow 3520=\frac{\theta}{360}\times2\times\frac{22}{7}\times6400\times\frac12.\]

Full parallel \(=\dfrac{44}{7}\times3200=20114.3\) km, so \(\theta=\dfrac{3520\times360}{20114.3}=63^{\circ}\) (eastward).

Starting at \(50^{\circ}\W\) and moving \(63^{\circ}\) east: \(-50^{\circ}+63^{\circ}=+13^{\circ}\). Longitude of \(L=13^{\circ}\E\).

(a)(ii) Latitude of N (due north of \(K\), along a meridian). Full meridian circle \(=2\pi R=\dfrac{44}{7}\times6400=40228.6\) km.

\[KN=\frac{\phi}{360}\times40228.6=10951\Rightarrow\phi=\frac{10951\times360}{40228.6}=98^{\circ}.\]

From \(60^{\circ}\N\), going \(98^{\circ}\) north first reaches the North Pole after \(30^{\circ}\), then continues \(68^{\circ}\) down the opposite meridian. Latitude of \(N=90^{\circ}-68^{\circ}=22^{\circ}\N\) (on the \(130^{\circ}\E\) meridian).

(b) Let total income be \(I\). Tax-free \(=20\%\), so taxable \(=0.8I\); tax \(=25\) kobo per naira \(=25\%\) of taxable:

\[0.25\times0.8I=1200\Rightarrow 0.2I=1200\Rightarrow I=\mathbf{N6000}.\]