Question

Peter was standing on the top of a rock cliff facing the sea. He saw a boat coming towards the shore. As he kept seeing time just flew. Ten minutes less than half of an hour, the angle of depression changed from 30 to 60. How much more time in minutes will the boat take to reach the shore ?

• A. 5
• B. 10
• C. 15
• D. 20

Answer 1

The Correct Option is B

To determine the additional time the boat will take to reach the shore, we first consider the problem's setup with trigonometry. A right triangle is formed with the cliff's height as one leg and the distance from the base of the cliff to the boat as the adjacent leg, changing as the boat approaches.

If the initial angle of depression is 30°, and later changes to 60°, we apply trigonometric relations. Let \( h \) be the height of the cliff and \( x_1 \) and \( x_2 \) be the initial and final distances from the base of the cliff to the boat when the corresponding angles of depression are 30° and 60°.

Using the tangent function:

$$ \tan(30°) = \frac{h}{x_1} $$ $$ \tan(60°) = \frac{h}{x_2} $$

From trigonometric properties: \( \tan(30°) = \frac{1}{\sqrt{3}} \) and \( \tan(60°) = \sqrt{3} \).

Equate and solve for distances:

$$ x_1 = \frac{h}{\tan(30°)} = h\sqrt{3} $$ $$ x_2 = \frac{h}{\tan(60°)} = \frac{h}{\sqrt{3}} $$

Since the angle changes from 30° to 60° in 10 minutes less than half of an hour (i.e., 20 minutes), we linearly relate distance covered over time.

The distance from \( x_1 \) to \( x_2 \) during this time period is:

$$ x_1 - x_2 = h\sqrt{3} - \frac{h}{\sqrt{3}} $$ $$ = \frac{3h - h}{\sqrt{3}} = \frac{2h}{\sqrt{3}} $$

If this distance corresponds to 20 minutes, and the remaining distance \( x_2 \) is:

$$ x_2 = \frac{h}{\sqrt{3}} $$

The rate of approach of the boat is the same throughout. Therefore, with a constant speed, the remaining time relates proportionally to the distance.

Thus, in another set of 20 minutes, the boat would double the distance covered: from \( x_1 \) to twice \( x_2 \), implying the remaining time to shore is:

$$ \frac{1}{2} \times 20 = 10 $$

The boat will take 10 more minutes to reach the shore.