Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problem 57

Chapter 6, Problem 57

Find the length and width of a rectangle whose perimeter is 36 feet and whose area is 77 square feet.

Verified step by step guidance

Let the length of the rectangle be $L$ feet and the width be $W$ feet.

Write the equation for the perimeter of the rectangle: \$2L + 2W = 36$.

Simplify the perimeter equation by dividing both sides by 2: $L + W = 18$.

Write the equation for the area of the rectangle: $L \times W = 77$.

Use the perimeter equation to express $L$ in terms of $W$: $L = 18 - W$, then substitute this into the area equation to get $(18 - W) \times W = 77$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around it, calculated by adding twice the length and twice the width (P = 2L + 2W). Understanding this formula allows you to set up an equation relating length and width when the perimeter is known.

Area of a Rectangle

The area of a rectangle is the amount of space inside it, found by multiplying the length by the width (A = L × W). This formula helps create a second equation when the area is given, which is essential for solving for both dimensions.

Solving Systems of Equations

When two equations involve two variables, such as length and width, solving the system means finding values that satisfy both simultaneously. Techniques include substitution or elimination, which are used here to find the rectangle’s dimensions from the perimeter and area equations.

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Related Practice

Textbook Question

Exercises 57–59 will help you prepare for the material covered in the next section. Subtract: $\frac{3}{x - 4}\) - \(\frac{2}{x + 2}$

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Textbook Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x²+y²<16, y≥2^x

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Textbook Question

Find the value of the objective function z = 2x + 3y at each corner of the graphed region shown. What is the maximum value of the objective function? What is the minimum value of the objective function?

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Textbook Question

In Exercises 57–59, graph the region determined by the constraints. Then find the maximum value of the given objective function, subject to the constraints. This is a piecewise function. Refer to the textbook.

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Textbook Question

Exercises 57–59 will help you prepare for the material covered in the next section. Add: (5x−3)/(x²+1) + 2x/(x²+1)².

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Textbook Question

Find the length and width of a rectangle whose perimeter is 40 feet and whose area is 96 square feet.

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