(a) The angle of depression of a boat from the... | WAEC SSCE - General Mathematics - 2010 (Essay)

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(a) The angle of depression of a boat from the mid-point of a vertical cliff is 35°. If the boat is 120m from the foot of the cliff, calculate the height of the cliff.

(b) Towns P and Q are x km apart. Two motorists set out at the same time from P to Q at steady speeds of 60 km/h and 80 km/h. The faster motorist got to Q 30 minutes earlier than the other. Find the value of x.

Answer

(a) The observer is at the mid-point of a vertical cliff. Let the mid-point be at height \(h\) above the foot, so the full cliff height is \(2h\). The boat is 120 m from the foot, and the angle of depression from the mid-point is \(35^\circ\).

The angle of depression equals the angle of elevation from the boat to the mid-point:

\[\tan 35^\circ = \frac{h}{120} \Rightarrow h = 120\tan 35^\circ = 120(0.7002) = 84.03\text{ m}.\]

Height of the cliff \(= 2h = 2(84.03) = 168\text{ m}\) (to 3 s.f.).

(b) Let the distance be \(x\) km. Time at 60 km/h is \(\dfrac{x}{60}\) h; time at 80 km/h is \(\dfrac{x}{80}\) h. The faster motorist arrives 30 minutes \(\left(\tfrac{1}{2}\text{ h}\right)\) earlier:

\[\frac{x}{60} - \frac{x}{80} = \frac{1}{2}.\]

\[x\left(\frac{4 - 3}{240}\right) = \frac{1}{2} \Rightarrow \frac{x}{240} = \frac{1}{2} \Rightarrow x = 120.\]

Therefore \(x = \mathbf{120\text{ km}}\).

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Answer Details

The angle of depression equals the angle of elevation from the boat to the mid-point:

\[\tan 35^\circ = \frac{h}{120} \Rightarrow h = 120\tan 35^\circ = 120(0.7002) = 84.03\text{ m}.\]

Height of the cliff \(= 2h = 2(84.03) = 168\text{ m}\) (to 3 s.f.).

\[\frac{x}{60} - \frac{x}{80} = \frac{1}{2}.\]

\[x\left(\frac{4 - 3}{240}\right) = \frac{1}{2} \Rightarrow \frac{x}{240} = \frac{1}{2} \Rightarrow x = 120.\]

Therefore \(x = \mathbf{120\text{ km}}\).

Read lesson note on Fractions, Decimals, Approximations And Percentage (JAMB)

Read lesson note on Progression (JAMB)

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(a) The angle of depression of a boat from the mid-point of a vertical cliff is 35°. If the boat is 120m from the foot of the cliff, calculate the height of...

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(a) The angle of depression of a boat from the mid-point of a vertical cliff is 35°. If the boat is 120m from the foot of the cliff, calculate the height of...

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