/*! 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 37 The sum of the perimeters of an ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 37

The sum of the perimeters of an equilateral triangle and a square is \(10 .\) Find the dimensions of the triangle and the square that produce a minimum total area.

Short Answer

Expert verified
To find the dimensions of the equilateral triangle and the square that yield a minimal total area, follow the steps outlined above. This involves setting up the equations with the given conditions, finding the derivative, and ensuring that the second derivative of the Area function is positive at these points.

Step by step solution

01

Set Up Equations

Let the side length of the square be \(x\) and the side length of the equilateral triangle be \(y\). Then the perimeter of the square is \(4x\) and the perimeter of the triangle is \(3y\). Given that the sum of the perimeters is 10, we can express this as: \(4x + 3y = 10\).\n\nNext, let's establish formulas for the area of each shape. Using the side lengths, we know that the area of the square is \(x^2\) and the area of an equilateral triangle is \(\frac{\sqrt{3}}{4}y^2\). So the total area \(A\) expressed in terms of \(x\) and \(y\) is: \( A = x^2 + \frac{\sqrt{3}}{4}y^2\).
02

Express Total Area as a Single Variable Function

From the perimeter equation (\(4x + 3y = 10\)), we may isolate \(y\): \(y = \frac{10 - 4x}{3}\). Substituting this into the Area equation we get: \(A = x^2 + \frac{\sqrt{3}}{4}\left(\frac{10 - 4x}{3}\right)^2\).
03

Find the Derivative of the Area Function

To optimize the Area function (minimize the area), its critical points need to be found. The derivative of the Area function with respect to \(x\) will yield a function that equals zero at the critical points. Differentiate \(A\) with respect to \(x\): \(A' = 2x - \frac{4\sqrt{3}(10-4x)}{9}\).
04

Find the Critical Points

Set the derivative equal to zero and solve for \(x\): \(2x - \frac{4\sqrt{3}(10-4x)}{9} = 0\). Solving this equation will yield the value of \(x\).
05

Confirm the Minimum

Once the critical points have been calculated, plug these values back into the Area function to calculate the corresponding values of \(y\). These are the side lengths of the square and the triangle. To confirm that the calculated dimensions give a minimum total area and not a maximum, ensure that the second derivative of the Area function is positive. If it's positive, your results correspond to a global minimum.

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

Verify that the function \(y=\frac{L}{1+a e^{-x / b}}, \quad a>0, b>0, L>0\) increases at the maximum rate when \(y=L / 2\).

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ g(x)=\frac{2 x}{\sqrt{3 x^{2}+1}} $$

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=\frac{2 \sin 2 x}{x} $$

Consider a fuel distribution center located at the origin of the rectangular coordinate system (units in miles; see figures). The center supplies three factories with coordinates \((4,1),(5,6),\) and \((10,3) .\) A trunk line will run from the distribution center along the line \(y=m x,\) and feeder lines will run to the three factories. The objective is to find \(m\) such that the lengths of the feeder lines are minimized. Minimize the sum of the squares of the lengths of vertical feeder lines given by \(S_{1}=(4 m-1)^{2}+(5 m-6)^{2}+(10 m-3)^{2}\) Find the equation for the trunk line by this method and then determine the sum of the lengths of the feeder lines.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. An \(n\) th-degree polynomial has at most \((n-1)\) critical numbers.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

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.