/*! 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

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?

A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of \(5 \mathrm{kg} / \mathrm{m}\). a. How much work is required to wind the entire chain onto the cylinder using the winch? b. How much work is required to wind the chain onto the cylinder if a \(50-\mathrm{kg}\) block is attached to the end of the chain?

Fill in the blanks: A region \(R\) is revolved about the \(y\) -axis. The volume of the resulting solid could (in principle) be found using the disk/washer method and integrating with respect to __________ or using the shell method and integrating with respect to __________.

Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. $$a(t)=\frac{20}{(t+2)^{2}}, v(0)=20, s(0)=10$$

On the first day of the year \((t=0),\) a city uses electricity at a rate of \(2000 \mathrm{MW}\). That rate is projected to increase at a rate of \(1.3 \%\) per year. a. Based on these figures, find an exponential growth function for the power (rate of electricity use) for the city. b. Find the total energy (in \(\mathrm{MW}\) -yr) used by the city over four full years beginning at \(t=0\). c. Find a function that gives the total energy used (in \(\mathrm{MW}\) -yr) between \(t=0\) and any future time \(t > 0\).
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

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.