The figure on the left above is a plan for a combination of a store and a park on a rectangular lot of width 30 feet. The length of the rectangular lot used will be x feet, the store will be a square portion of the lot with sides of length (2/3}x feet, and the rest of the 30 foot by x foot lot will be the park. The area of the park will then be A square feet, where A(x) = 30x -( 2/3}x )^2. The graph of A(x) is shown on the right. The designer of the store-park combination wants the park to be at least 400 square feet in area. Of the following, which is closest to the shortest length of the lot that will produce a park of this area?
18 feet is closest to the shortest length of the lot that will produce a park of at least 400 square feet in area.
To determine the shortest length of the lot, we need to find the value of x for which the area of the park A(x) is at least 400 square feet. By solving the equation \( 30x - \left(\frac{2}{3}x\right)^2 \geq 400 \), we find that the smallest x satisfying this condition is approximately 18 feet.
If the length of the lot is 13 feet, we can calculate the area of the park as follows:
\[
A(13) = 30(13) - \left(\frac{2}{3}(13)\right)^2 = 390 - \left(\frac{26}{3}\right)^2 \approx 390 - 227.56 \approx 162.44 \text{ square feet}.
\]
This area is well below 400 square feet, making this choice incorrect.
For a length of 18 feet, we calculate the area of the park:
\[
A(18) = 30(18) - \left(\frac{2}{3}(18)\right)^2 = 540 - \left(12\right)^2 = 540 - 144 = 396 \text{ square feet}.
\]
This area is just under 400 square feet, making it the closest option to meet the requirement.
If the length is 33 feet, we find the area as follows:
\[
A(33) = 30(33) - \left(\frac{2}{3}(33)\right)^2 = 990 - \left(22\right)^2 = 990 - 484 = 506 \text{ square feet}.
\]
While this option meets the area requirement, it is longer than necessary.
Calculating the area for 49 feet gives us:
\[
A(49) = 30(49) - \left(\frac{2}{3}(49)\right)^2 = 1470 - \left(32.67\right)^2 \approx 1470 - 1067.11 \approx 402.89 \text{ square feet}.
\]
This area also exceeds 400 square feet, making this choice even longer than necessary.
The goal is to find the shortest length of the lot that allows for a park area of at least 400 square feet. Among the options provided, 18 feet yields an area of 396 square feet, which is closest to the requirement without exceeding it. Hence, 18 feet
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