Daily Regents: Rectangular Garden

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

June 2014, Questions 34

34. A rectangular garden measuring 12 meters by 16 meters is to have a walkway installed around it with a width of x meters, as shown in the diagram below. Together the walkway and the garden have an area of 396 square meters.

Write an equation that can be used to find x, the width of the walkway.

Describe how your equation models the situation.

Determine and state the width of the walkway, in meters.

Unlike the previous question, this was a majorly complicated problem, with more binomial multiplication.

The length of the rectangle is x + 16 + x or (2x + 16).
The width of the rectangle is x + 12 + x or (2x + 12).
The area of the rectangle is 396.

So (2x + 16)(2x + 12) = 396
4x² + 24x + 32x + 192 = 396
4x² + 56x - 204 = 0
Any time you have a coefficient in front of x², check if you can factor it out. In this case, all the coefficients are multiples of 4, so we can just divide each term on the left by 4. Dividing 0 by 4 will leave 0 on the right side.
x² + 14x - 51 = 0 Factor the trinomial back into two binomials and apply the Zero Product Property. (One of the binomials MUST equal zero.)
(x + 17)(x - 3) = 0
x + 17 = 0 or x - 3 = 0
x = -17 or x = 3

We can discard the negative value not because we made a mistake, but because a negative value makes no sense in the context of length and width.
You are left with the width of the walkway being 3 meters wide.

The equation models the situation because the Area is the product of the length times the width. The length of the garden plus the walkway on either side is 2x + 16. The width of the garden plus the walkway is 2x + 12. The Area of the garden is 396 square meters.

Any questions?

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