/*! 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 73 A rectangular playing field with... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 73

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 be between \(25 - \sqrt{225}\) and \(25 + \sqrt{225}\) meters.

Step by step solution

01

Formulate the perimeter

The formula for the perimeter of a rectangle is \( P = 2 \times (length + width) \). Given that the perimeter is 100, we can say: \( 2 \times (length + width) = 100 \) or \( length + width = 50 \).
02

Solve for the width

From step 1, we can rearrange the formula to solve for the width. \( width = 50 - length \)
03

Formulate the area

The area of a rectangle is length 脳 width. With the width formula from Step 2, the area becomes: \( Area = length \times (50 - length) \)
04

Solve for the length

Given that the area is at least 500, we can set up an inequality: \( 500 \leq length \times (50 - length) \). Solving this quadratic inequality gives the bounds for the length of the rectangle.
05

Solve the inequality

By solving the inequality we get two roots, these roots show us the bounds. It is found that \(25 - \sqrt{225} \leq length \leq 25 + \sqrt{225}\). So the length of the rectangle must be within these bounds

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

Find a polynomial function that has the given zeros. 0,-7

Use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\), where \(s\) represents the height of an object (in feet), \(v_{0}\) represents the initial velocity of the object (in feet per second), \(s_{0}\) represents the initial height of the object (in feet), and \(t\) represents the time (in seconds). A projectile is fired straight upward from ground level \(\left(s_{0}=0\right)\) with an initial velocity of 160 feet per second. (a) At what instant will it be back at ground level? (b) When will the height exceed 384 feet?

Between two consecutive zeros, a polynomial must be entirely ______________ or entirely __________________.

(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. \(3 x^{2}+b x+10=0\)

(a)verify the given factors of the function f,(b) find the remaining factor(s) of f,(c) use your results to write the complete factorization of f,(d) list all real zeros of f, and (e) confirm your results by using a graphing utility to graph the function. \(f(x)=6 x^{3}+41 x^{2}-9 x-14 \quad(2 x+1),(3 x-2)\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

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.