(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.
Explanation
(a)
(frac{FM}{FB} = tan 35)
(therefore frac{FM}{120} = tan 35)
(FM = 120 tan 35 )
= (120 times 0.7002)
= (84.024 m)
Therefore, the height of the cliff = (2 times 84.024 = 168.048m)
(b) Distance between P and Q = x km
Speed for 1st motorist = 60 km/h
(therefore text{1st motorist’s time} = frac{x}{60} hrs)
Speed for 2nd motorist = 80 km/h
(therefore text{2nd motorist’s time} = frac{x}{80} hrs)
(frac{x}{60} – frac{x}{80} = frac{30}{60})
(frac{4x – 3x}{240} = frac{1}{2})
(frac{x}{240} = frac{1}{2} implies 2x = 240)
(x = 120 km).