/*! 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 75 A rectangular playing field with... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 75

A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters. Within what bounds must the length of the rectangle lie?

Short Answer

Expert verified
The length of the rectangle must lie between 50 meters and the positive root of the quadratic equation evaluated in Step 3.

Step by step solution

01

Understand the Problem

We need to calculate the length of the rectangle, given its perimeter and minimum area. We know that 2 * (Length + Width) = Perimeter and Length * Width = Area. These are our two primary formulas for this problem.
02

Set up the Equations

From the Perimeter formula, we know that 2 * (Length + Width) = 100, so Width = 50 - Length. Using this equation, we can express the Area formula in terms of Length: Length * (50 - Length) >= 500. The inequality indicates that the area must be at least 500.
03

Find the Roots of the Quadratic Equation

From the equation derived in Step 2, we see that it is a quadratic inequality. Thumbsolve it to get Length^2 - 50Length + 500 = 0. This can be solved using the quadratic formula Length = [-(-50) 卤 sqrt((50)^2 - 4*1*500)] / 2*1. This yields two results that give the lower and upper limit of the length.
04

Interpret the Results

The negative root of the quadratic equation doesn't make sense in this context, because the length can't be negative. So, discard the negative root. The other root gives the maximum length. The minimum Length is half of the perimeter, i.e., 50 meters. So, that's the range for the length of the rectangle.

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 statement is true or false. Justify your answer. The solution set of the inequality \(\frac{3}{2} x^{2}+3 x+6 \geq 0\) is the entire set of real numbers.

Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{1}{4} x^{2}-x+\frac{1}{4}=\frac{1}{4}\left(4 x^{3}-x^{2}-4 x+1\right)$$

Use the given zero to find all the zeros of the function. Function \(f(x)=x^{4}+3 x^{3}-5 x^{2}-21 x+22\) Zero \(-3+\sqrt{2} i\)

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$2 x^{2}+b x+5=0$$

Think About It \(\quad\) A cubic polynomial function \(f\) has real zeros \(-2, \frac{1}{2},\) and \(3,\) and its leading coefficient is negative. Write an equation for \(f\) and sketch its graph. How many different polynomial functions are possible for \(f ?\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.