Question

Tina is planning to fence in an area for her dog. She wants to create a rectangular area of 1,600 square feet for her pet and can afford to purchase 160 feet of fence. In two or more complete sentences, explain the algebraic model, calculations and reasoning necessary to determine the dimensions of the rectangular area.

Answer 1

Using appropriate formulas, the dimensions of a rectangle (length and width) can be used to calculate the area of the rectangle and its perimeter. Using the given information, those formulas give two equations in the two unknowns of length and width. Using any of a variety of methods, the solution that satisfies both equations can be found. That solution will provide numerical values for the length and width of the area for the dog.

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A = 1600 = L·W

P = 160 = 2(L + W)

Solving the second for L and substituting into the first, we get

L = 80 - W

1600 = (80 -W)·W

W² -80W +1600 = 0 . . . . . subtract the right side, put in standard form

(W -40)² = 0 . . . . . . . . . . . . factor the equation

W = 40 . . . . . . . . . . . . . . . . this has one solution, of multiplicity 2

L = 80 - 40 = 40 . . . . . . . . use the previously developed equation to find L

The dog pen will be a 40-foot square.

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The graph shows another way to solve these equations. The solution is the point(s) where the curves intersect. There is only one point: L = W = 40.

Answer 2

First of all, since A = l x w for rectangles, and A = 1600 ft ^ 2, you could do the square root of 1600 and get 40.

This means that 40 x 40 = 1600.

So the length = 40 and the width = 40.

Also, since Tina can get 160 feet of fence, which is the perimeter, you could do 40 + 40 + 40 + 40 = 160, which confirms that the length = 40 and the width = 40.