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Chapter 13: 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 rectangle has dimensions 11 feet by 7 feet.

Step by step solution

01

Formulate the equations

From the problem, we're given that the perimeter of the rectangle is 36 feet and the area is 77 square feet. This gives us two equations: \(2(l + w) = 36\) and \(lw = 77\).
02

Simplify the perimeter equation

Divide the perimeter equation by 2 to simplify it, which gives us \(l + w = 18\). Let's call this equation (1).
03

Formulate another equation in terms of one variable

Rearrange equation (1) to find \(l = 18 - w\). This is equation (2).
04

Substitute equation (2) into the area equation

Substitute equation (2) into the area equation: \((18 - w)w = 77\). Expand and rearrange this to get a quadratic equation in terms of \(w\), which is \(w^2 -18w + 77 = 0\).
05

Solve the quadratic equation

Solve the quadratic equation to get the possible values for \(w\). The solutions are \(w = 7\) and \(w = 11\). These are the possible widths of the rectangle in feet.
06

Find the corresponding length for each width

Substitute each \(w\) in equation (2) to get the corresponding length \(l\). For \(w = 7\), \(l = 18 - 7 = 11\). For \(w = 11\), \(l = 18 - 11 = 7\).
07

Conclude the solution

The rectangle has a length of 11 feet and width of 7 feet or vice versa. The two answers correspond to the same rectangle, just oriented differently.

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Most popular questions from this chapter

Graph: \(f(x)=2^{1-x}\). (Section \(12.1,\) Example 4 )

Solve the systems. $$\left\\{\begin{array}{l} \log _{y} x=3 \\ \log _{y}(4 x)=5 \end{array}\right.$$

Will help you prepare for the material covered in the next section. Solve by the substitution method: $$\left\\{\begin{array}{c}4 x+3 y=4 \\\y=2 x-7\end{array}\right.$$

What is a parabola?

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=\frac{1}{2}(y+2)^{2}+1$$
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