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Chapter 2: Problem 122

A boat travels from south bank to north bank of river with a maximum speed of \(8 \mathrm{~km} / \mathrm{h}\). A river current flows from west to east with a speed of \(4 \mathrm{~km} / \mathrm{h}\). To arrive at a point opposite to the point of start, the boat should start at an angle (A) \(\tan ^{-1}(1 / 2)\) west of north (B) \(\tan ^{-1}(1 / 2)\) north of west (C) \(30^{\circ}\) west of north (D) \(30^{\circ}\) north of west

Short Answer

Expert verified
The boat should start at an angle of \(30^{\circ}\) west of north to counteract the river current and arrive at a point opposite to its starting point.

Step by step solution

01

Understand the problem context

The boat is traveling from the south bank to the north bank. A river current is flowing from west to east with a speed of 4 km/h. The goal is to find the angle at which the boat should start its journey in order to end up directly opposite the starting point on the north bank. This means we will need to find a way to counteract the effect of the current on the movement of the boat. 2.
02

Split the boat velocity

As the boat has to counteract the effect of the current we will have to split the boat velocity into two components 鈥� the component in the north direction and the component in the west direction. We can use the following equations: \(v_{north} = v_{boat} \times \cos{\theta}\) \(v_{west} = v_{boat} \times \sin{\theta}\) Here, \(v_{boat}\) is the maximum speed of the boat, which is 8 km/h, and \(\theta\) is the angle we need to find. 3.
03

Counteract the river current

In order to counteract the river current, the westward velocity of the boat needs to be equal and opposite to the river's speed. So, we can equate the westward velocity of the boat to the river current speed: \(v_{west} = v_{current}\) \(v_{boat} \times \sin{\theta} = 4\) 4.
04

Find the sine of the angle

From the previous step, we have the relation between the speed of the boat and sin胃. We can find sin胃: \(\sin(\theta) = \frac{v_{current}}{v_{boat}}\) \(\sin(\theta) = \frac{4}{8}\) \(\sin(\theta) = 0.5\) 5.
05

Find the angle 胃

Now that we have found the sine of the angle, we can find the angle 胃 in degrees: \(\theta = \sin^{-1}(0.5)\) \(\theta = 30^{\circ}\) Since the angle is measured from the north direction towards the west, the angle is 30掳 west of north, which corresponds to answer (C).

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Most popular questions from this chapter

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