/*! 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 48 An electronic sign for a grocery... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 48

An electronic sign for a grocery store is in the shape of a rectangle. The perimeter of the sign is \(72 \mathrm{ft}\) and the area is \(320 \mathrm{ft}^{2}\). Find the length and width of the sign.

Short Answer

Expert verified
The length is 20 ft and the width is 16 ft.

Step by step solution

01

Identify the Given Information

The perimeter of the rectangle is given as 72 ft and the area is given as 320 ft虏.
02

Simplify the Perimeter Equation

Divide the entire perimeter equation by 2: \[ 36 = l + w \]
03

Set Up the Area Equation

The formula for the area of a rectangle is: \[ A = lw \] Using the given area, 320 ft虏: \[ 320 = lw \]
04

Express One Variable in Terms of the Other

From the simplified perimeter equation \( 36 = l + w \), express the width in terms of the length: \[ w = 36 - l \]
05

Substitute into the Area Equation

Substitute \( w = 36 - l \) into the area equation: \[320 = l(36 - l) \] Simplify to get a quadratic equation: \[320 = 36l - l^2 \] \[ l^2 - 36l + 320 = 0 \]
06

Solve the Quadratic Equation

Solve the quadratic equation \( l^2 - 36l + 320 = 0 \) using the quadratic formula \( l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \(a = 1\), \(b = -36\), and \(c = 320\): \[ l = \frac{36 \pm \sqrt{1296 - 1280}}{2} \] \[ l = \frac{36 \pm \sqrt{16}}{2} \] \[ l = \frac{36 \pm 4}{2} \] Solve for the two possible roots: \[ l = \frac{40}{2} = 20 \] and \[ l = \frac{32}{2} = 16 \]
07

Determine the Dimensions

Using the two possible values for the length, calculate the corresponding widths: For \( l = 20 \), \[ w = 36 - 20 = 16 \] For \( l = 16 \), \[ w = 36 - 16 = 20 \] Therefore, the dimensions of the rectangle are 20 ft by 16 ft.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perimeter of Rectangles
Understanding the perimeter of rectangles is crucial for solving many geometric problems. The perimeter of a rectangle is the total distance around its outer edge. You calculate it by adding the lengths of all four sides. In simpler terms, this can be done using the formula:

\[ P = 2l + 2w \]
Where:
  • P is the perimeter,
  • l is the length of the rectangle,
  • w is the width of the rectangle.
Dividing the entire perimeter equation by 2 simplifies the equation:
\[ \frac{P}{2} = l + w \]
This simplified form helps in setting up equations for solving unknown dimensions when the perimeter is given, making it easier to find the length and width.
Area of Rectangles
Next, we tackle the concept of the area of rectangles. The area represents the total space covered by the rectangle. You calculate the area with the formula:

\[ A = lw \]
Where:
  • A is the area,
  • l is the length,
  • w is the width.
In our example, the area was given as 320 square feet. To find the dimensions from the area, you rearrange the formula to solve for one variable in terms of the other. By substituting one variable into this equation, you can form a quadratic equation, which brings us to our next concept.
Quadratic Formula
The quadratic formula is essential for solving equations of the form:

\[ ax^2 + bx + c = 0 \]
In this equation,
  • a, b, and c are constants,
  • x represents the variable you need to solve for.
The quadratic formula to find the values of x is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our solution, the quadratic equation derived from the perimeter and area equations was:

\[ l^2 - 36l + 320 = 0 \]
Using the quadratic formula with:
  • a = 1,
  • b = -36,
  • c = 320,
we found that:
\[ l = \frac{36 \pm \sqrt{1296 - 1280}}{2} = \frac{36 \pm 4}{2} \]
This yielded the two possible values for the length: 20 and 16. With these lengths, corresponding widths were found to be 16 and 20 respectively. The quadratic formula thus facilitated finding the precise dimensions of the rectangle.

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

A manufacturer produces two models of a gas grill. Grill A requires 1 hr for assembly and \(0.4 \mathrm{hr}\) for packaging. Grill \(B\) requires 1.2 hr for assembly and 0.6 hr for packaging. The production information and profit for each grill are given in the table. (See Example 4\()\) $$ \begin{array}{|l|c|c|c|} \hline & \text { Assembly } & \text { Packaging } & \text { Profit } \\ \hline \text { Grill A } & 1 \mathrm{hr} & 0.4 \mathrm{hr} & \$ 90 \\ \hline \text { Grill B } & 1.2 \mathrm{hr} & 0.6 \mathrm{hr} & \$ 120 \\ \hline \end{array} $$ The manufacturer has \(1200 \mathrm{hr}\) of labor available for assembly and \(540 \mathrm{hr}\) of labor available for packaging. a. Determine the number of grill A units and the number of grill B units that should be produced to maximize profit assuming that all grills will be sold. b. What is the maximum profit under these constraints? c. If the profit on grill A units is $$\$ 110$$ and the profit on grill \(\underline{B}\) units is unchanged, how many of each type of grill unit should the manufacturer produce to maximize profit?

Solve the system using any method. $$ \begin{array}{l} y=-0.18 x+0.129 \\ y=-0.15 x+0.1275 \end{array} $$

Solve the system using any method. $$ \begin{array}{l} 3(2 x-y)=2-x \\ x+\frac{5}{4} y=\frac{3}{2} \end{array} $$

The population \(P(t)\) of a culture of bacteria grows exponentially for the first 72 hr according to the model \(P(t)=P_{0} e^{k t} .\) The variable \(t\) is the time in hours since the culture is started. The population of bacteria is 60,000 after 7 hr. The population grows to 80,000 after 12 hr. a. Determine the constant \(k\) to 3 decimal places. b. Determine the original population \(P_{0}\). Round to the nearest thousand. c. Determine the time required for the population to reach 300,000 . Round to the nearest hour.

Solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples \(5-6\) ) $$ \begin{array}{r} 3 x+y=6 \\ x+\frac{1}{3} y=2 \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

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.