A rectangle has dimensions of length (L) and width (W). The area is defined as L * W, so we have L * W = 27. On the other hand, the perimeter of a rectangle is the sum of lengths of its sides: that is L + W + L + W = 2 * L + 2 * W. From the problem statement, 2 * L + 2 * W = 24. Factor the left-hand side of this equation to get
2 *(L + W) = 24, then divide both sides by 2, getting L + W = 12. So, we know L + W = 12 and L * W = 27.
Now we have two equations in two unknowns: hard to solve. Approach? Express one of the two variables in terms of the other. Say, for example, express L in terms of W: If L + W = 12, then L = 12 - W. Now we have
L * W = 27 means that (12 - W) * W = 27. Multiply out the left-hand side, getting 12 * W - W^2 = 27. Bringing everything over to the right-hand side of the equals sign, we get 27 + W^2 - 12 * W = 0. Rewriting this in standard form, we have W^2 - 12 * W + 27 = 0. This can be factored as (W - 3) * (W - 9) = 0, which has W = 3 and W = 9 as solutions. If W = 3 then L = 9 and if W = 9, then W = 9 and L = 3. Usually we think of L > W, so L = 9 and W = 3 is the solution. Check: L * W = 9 * 3 = 27. Check L + W = 3 + 9 = 12.
May 4th, 2015
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