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Chapter 8: Problem 57

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

Short Answer

Expert verified
The dimensions of the rectangle are \(L = 11\) feet and \(W = 7\) feet.

Step by step solution

01

Use the Given Perimeter to Form an Equation

From the given equation, we can derive an equation relating the length \(L\) and width \(W\) of the rectangle from the perimeter formula. The formula to calculate the perimeter of a rectangle is \(2L + 2W\). Replacing 36 feet, this becomes: \(2L + 2W = 36\). Re-arranging this equation, we have: \(L = 18 - W\).
02

Use the Given Area to Form a Second Equation

Similarly using the formula for the area of a rectangle which is \(L \times W\), and substituting the given 77 square feet area, we can form another equation: \(L \times W = 77\). Now substitute the value of \(L\) from first step into this equation, we have: \((18 - W) \times W = 77\).
03

Solve the Quadratic Equation

Rearranging the equation \((18 - W) \times W = 77\) gives us a quadratic equation: \(W^2 - 18W + 77 = 0\). By solving this equation, we can find the possible values for the width \(W\).
04

Solve for Length

Once we have the width \(W\), we can substitute this into the equation from Step 1, \(L = 18 - W\), to find the value for length \(L\).

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Most popular questions from this chapter

Graphing urilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in rwo variables Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing urility to graph the inequalities in Exercises \(97-102\). $$3 x-2 y \geq 6$$

What is a constraint in a linear programming problem? How is a constraint represented?

determine whether each statement makes sense or does not make sense, and explain your reasoning. Because \(x+5\) is linear and \(x^{2}-3 x+2\) is quadratic, I set up the following partial fraction decomposition: $$\frac{7 x^{2}+9 x+3}{(x+5)\left(x^{2}-3 x+2\right)}=\frac{A}{x+5}+\frac{B x+C}{x^{2}-3 x+2}$$

determine whether each statement makes sense or does not make sense, and explain your reasoning. Because \((x+3)^{2}\) consists of two factors of \(x+3,1\) set up the following partial fraction decomposition: $$\frac{5 x+2}{(x+3)^{2}}=\frac{A}{x+3}+\frac{B}{x+3}$$

Solve: \(\sqrt{2 x-5}-\sqrt{x-3}=1 .\) (Section \(1.6,\) Example 4 )
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