/*! 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 118 A boat which has a speed of \(5 ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 118

A boat which has a speed of \(5 \mathrm{~m} / \mathrm{s}\) in still water crosses the river of width \(25 \mathrm{~m}\) in \(10 \mathrm{~s}\). The boat is heading at an angle of \(\alpha\) with downstream, where \(\alpha\) is equal to (A) \(150^{\circ}\) (B) \(120^{\circ}\) (C) \(90^{\circ}\) (D) \(60^{\circ}\)

Short Answer

Expert verified
The angle 伪 the boat is heading with downstream is \(60^{\circ}\).

Step by step solution

01

Identify given quantities

In this problem, we are given: 1. Speed of the boat in still water: \(v_b = 5 \mathrm{~m/s}\) 2. Width of the river: \(d = 25 \mathrm{~m}\) 3. Time taken to cross the river: \(t = 10 \mathrm{~s}\) 4. We need to find the angle 伪 the boat is heading with downstream.
02

Calculate the boat's speed across the river

To do this, we can use the formula: \(distance = speed \cdot time \) Let the boat's speed across the river be \(v_x\). First, we calculate this using the given width of the river and time taken to cross the river. \[v_x = \dfrac{d}{t} = \dfrac{25\mathrm{~m}}{10\mathrm{~s}} = 2.5 \mathrm{~m/s} \]
03

Calculate the boat's downstream speed

Since we know the boat's speed in still water (\(v_b\)) and its speed across the river (\(v_x\)), we can now find the boat's downstream speed, denoted by \(v_y\). To do this, we can use the Pythagorean theorem, as the boat's velocity components form a right-angled triangle. \[v_b^2 = v_x^2 + v_y^2 \] Now we can solve for \(v_y\): \[v_y = \sqrt{v_b^2 - v_x^2} = \sqrt{(5\mathrm{~m/s})^2 - (2.5\mathrm{~m/s})^2} = 4.33 \mathrm{~m/s} \]
04

Calculate the angle 伪 using the trigonometric identity

Now we have both components of the boat's velocity (\(v_x\) and \(v_y\)). We can determine the angle 伪 by using the tangent function. \[\tan(\alpha) = \frac{v_y}{v_x}\] Solving for the angle 伪: \[\alpha = \tan^{-1} \left(\dfrac{v_y}{v_x}\right) = \tan^{-1} \left(\dfrac{4.33 \mathrm{~m/s}}{2.5 \mathrm{~m/s}}\right) = 60^{\circ}\] So, the correct answer is: (D) \(60^{\circ}\)

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