Final answer:
To find the possible widths of the lot, we set up the perimeter and area equations for a rectangle, solve for one variable, substitute into the other equation, and solve the resulting quadratic inequality.
Step-by-step explanation:
The student asked for help in writing and solving an inequality to describe the possible widths of an oceanfront lot with a specific perimeter and a minimum area. Given a perimeter of 250 feet and a minimum area of 2500 square feet, we will assume that the length is l and the width is w. The perimeter (P) of a rectangle is calculated as P = 2l + 2w and the area (A) as A = lw. Therefore:
P = 2l + 2w = 250
A = lw ≥ 2500
Let's solve for one of the variables. We can express l in terms of w from the perimeter equation:
l = (250 - 2w) / 2
Then, substituting l into the area equation:
(250 - 2w)/2 ⋅ w ≥ 2500
(250w - 2w^2) ≥ 5000
-2w^2 + 250w - 5000 ≥ 0
Divide each term by -2 to reverse the inequality:
w^2 - 125w + 2500 ≤ 0
To find the possible values of w, we need to determine the roots of the quadratic equation w^2 - 125w + 2500 = 0. Upon finding the roots, the values of w between (and including) these roots will satisfy the inequality, denoting the possible widths of the lot.