/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 119 A swimmer wishes to cross a \(80... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 119

A swimmer wishes to cross a \(800 \mathrm{~m}\) wide river flowing at \(6 \mathrm{~km} / \mathrm{hr}\). His speed with respect to water is \(4 \mathrm{~km} / \mathrm{hr}\). He crosses the river in shortest possible time. He is drifted downstream on reaching the other bank by a distance of (A) \(800 \mathrm{~m}\) (B) \(1200 \mathrm{~m}\) (C) \(400 \sqrt{13} \mathrm{~m}\) (D) \(2000 \mathrm{~m}\)

Short Answer

Expert verified
The swimmer is drifted downstream by 1200 meters on reaching the other bank. Therefore, the correct answer is (B) \(1200 \mathrm{~m}\).

Step by step solution

01

Convert speeds to the same units

Since the width of the river is given in meters, let's convert the swimmer's speed and the river's speed to meters per second (m/s). Speed of river flow: \(\frac{6000 \mathrm m}{3600 \mathrm s} = \frac{5}{3}\mathrm{ms}^{-1}\) Speed of swimmer wrt water: \(\frac{4000 \mathrm m}{3600 \mathrm s} = \frac{10}{9}\mathrm{ms}^{-1}\) Now we have the speed of the river as \(\frac{5}{3}\mathrm{ms}^{-1}\) and the speed of the swimmer wrt water as \(\frac{10}{9}\mathrm{ms}^{-1}\).
02

Find the swimmer's optimal direction

For the swimmer to cross the river in the shortest possible time, he should swim perpendicular to the current. That means the swimmer's speed perpendicular to the river flow is \(\frac{10}{9}\mathrm{ms}^{-1}\) and the same speed can be considered as the resultant speed for the shortest crossing time.
03

Calculate the time taken to cross the river

Now, we will find the time taken to cross the river. We have the swimmer's resultant speed and the river's width, so we can use the formula: \[time = \frac{distance}{speed}\] \(time = \frac{800\mathrm{m}}{\frac{10}{9}\mathrm{ms}^{-1}}= 720\mathrm{s}\) So the swimmer takes 720 seconds to cross the river.
04

Calculate the downstream distance

To find the distance the swimmer is drifted downstream, we can use the river's speed and the time taken to cross the river: Downstream distance = River flow speed * Crossing time Downstream distance = \(\frac{5}{3}\mathrm{ms}^{-1} * 720\mathrm{s} = 1200\mathrm{m}\) The swimmer is drifted downstream by 1200 meters on reaching the other bank. Therefore, the correct answer is: (B) \(1200 \mathrm{~m}\)

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Most popular questions from this chapter

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