Question

Jane bought a new rectangular placemat with an area of 126 square inches. The length of the placemat is four times the quantity of nine less than half its width. Complete the equation that models the area of the placemat, in terms of the width of the placemat, w. = w2 - w The width of the placemat is inches.

Answer 1

The area of the place mat, 126 in^2, is equal to L * W, where L = 4(W/2 - 9).

We must solve this equation for the width, W.

Then 126 in^2 = W * 4(w/2 - 9).  Simplifying, 126 in^2 =  W(2W - 36), and

63 in^2 = W(W - 18).  Then W^2 - 18W - 63 = 0.  This is a quadratic equation with coefficients a = 1, b = -18 and c = -63; the discriminant is 576.

Thus, the solutions are:

         -(-18) plus or minus √576

W = -----------------------------------------

                         2

          18 plus or minus 24

    = --------------------------------- = {21, -3}.  Omit the negative result, since

                        2                        W = width must be a positive quantity.

Thus, W = 21 inches.

Answer 2

The width of the placemat with an area of 126 square inches is 21 inches.

What is an equation?

An equation is an expression that shows the relationship between two or more numbers and variables.

Let w represent the width of the placemat. The length is four times the quantity of nine less than half its width.

Length = 4(w/2 - 9) = 2w - 36

Area = length * width

126 = w(2w - 36)

2w² - 36w - 126 = 0

w = 21

The width of the placemat with an area of 126 square inches is 21 inches.

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