/*! 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 61 For each region \(R\), find the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 61

For each region \(R\), find the horizontal line \(y=k\) that divides \(R\) into two subregions of equal area. \(R\) is the region bounded by \(y=1-|x-1|\) and the \(x\) -axis.

Short Answer

Expert verified
#tag_content# The value of k for the horizontal line y=k that divides the region R into two equal parts is found by following these steps: 1. Identify intersection points with the x-axis by setting y=0 and solving for x. 2. Calculate the total area under the curve using an integral. 3. Determine the x-value c that corresponds to half the area by solving an integral equation. 4. Find the k-value by substituting the x-value c back into the equation y=1-|x-1|. By following these steps, the value of k can be determined.

Step by step solution

01

Determine where the curve intersects the x-axis.

To find the intersection points, set \(y\) equal to \(0\) and solve for \(x\): $$0 = 1 - |x-1|$$ Solve for \(x\) when \(x-1 \ge 0\) and when \(x-1 < 0\) separately to obtain the intersection points. #Step 2: Calculate the total area under the curve#
02

Set up and evaluate an integral to find the area under the curve.

The total area under the curve is given by the integral: $$A = \int_{a}^{b} (1 - |x-1|) dx$$ where \(a\) and \(b\) are the intersection points with the x-axis found in step 1. Calculate the area by splitting the integral into two parts based on the absolute value function. #Step 3: Determine the half-area line and the k-value#
03

Find the line \(y=k\) that divides the region into two equal areas.

Since we want to find the horizontal line that divides the region into two equal areas, we can set the two equal areas as: $$\frac{A}{2} = \int_{a}^{c} (1-|x-1|) dx$$ where \(c\) is a value between the intersection point \(a\) and \(b\). Solve for the x-value \(c\) and substitute back into the equation \(y=1-|x-1|\) to find \(k\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Determine whether the following statements are true and give an explanation or counterexample. a. When using the shell method, the axis of the cylindrical shells is parallel to the axis of revolution. b. If a region is revolved about the \(y\) -axis, then the shell method must be used. c. If a region is revolved about the \(x\) -axis, then in principle, it is possible to use the disk/washer method and integrate with respect to \(x\) or the shell method and integrate with respect to \(y\)

Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in \(\mathrm{mi} / \mathrm{hr}\) ). Assume \(t\) is measured in hours. Theo: \(v_{T}(t)=10,\) for \(t \geq 0\) Sasha: \(v_{S}(t)=15 t,\) for \(0 \leq t \leq 1\) and \(v_{S}(t)=15,\) for \(t>1\) a. Graph the velocity functions for both riders. b. If the riders ride for 1 hr, who rides farther? Interpret your answer geometrically using the graphs of part (a). c. If the riders ride for 2 hr, who rides farther? Interpret your answer geometrically using the graphs of part (a). d. Which rider arrives first at the \(10-, 15-\), and 20 -mile markers of the race? Interpret your answer geometrically using the graphs of part (a). e. Suppose Sasha gives Theo a head start of \(0.2 \mathrm{mi}\) and the riders ride for 20 mi. Who wins the race? f. Suppose Sasha gives Theo a head start of \(0.2 \mathrm{hr}\) and the riders ride for 20 mi. Who wins the race?

Find the area of the region bounded by the curve \(x=\frac{1}{2 y}-\sqrt{\frac{1}{4 y^{2}}-1}\) and the line \(x=1\) in the first quadrant. (Hint: Express \(y\) in terms of \(x\).)

Find the volume of the solid of revolution. Sketch the region in question. The region bounded by \(y=1 / \sqrt{x}, y=0, x=2,\) and \(x=6\) revolved about the \(x\) -axis

City A has a current population of 500,000 people and grows at a rate of \(3 \% /\) yr. City \(\mathrm{B}\) has a current population of 300,000 and grows at a rate of \(5 \% / \mathrm{yr}\). a. When will the cities have the same population? b. Suppose City C has a current population of \(y_{0} < 500,000\) and a growth rate of \(p>3 \% /\) yr. What is the relationship between \(y_{0}\) and \(p\) such that the Cities \(A\) and \(C\) have the same population in 10 years?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.