Question

A rectangular parking lot must have a perimeter of 560 feet and an area of at least 10,000 square feet. Describe the possible lengths of the parking lot.

Answer 1

The length of the parking lot can be either 10 feet or 270 feet.

Let L be the length and W be the width of the parking lot. We are given that the perimeter is 560 feet, so we can write the equation:

2L + 2W = 560

We are also given that the area must be at least 10,000 square feet, so we can write the inequality:

LW ≥ 10,000

We can solve for L in terms of W from the first equation:

L = 280 - W

Substituting this expression for L into the second inequality, we get:

(280 - W)W ≥ 10,000

Expanding and rearranging, we get:

W^2 - 280W + 10,000 ≤ 0

Factoring, we get:

(W - 10)(W - 270) ≤ 0

Therefore, the possible values for W are 10 and 270. Substituting these values back into the equation for L, we get the possible values for L:

L = 270 and L = 10

Therefore, the possible lengths of the parking lot are 10 feet and 270 feet.

Answer 2

To find the possible lengths of the parking lot, we use the formulas for perimeter and area of a rectangle. By solving an inequality involving the area and using the quadratic formula, we find the possible lengths to be 10 feet and 270 feet.

Step-by-step explanation:

To find the possible lengths of the rectangular parking lot, we need to work with the given information. Let's denote the length of the parking lot as L and the width as W. The formula for the perimeter of a rectangle is P = 2L + 2W, so we have 2L + 2W = 560. Rearranging this equation, we get L = 280 - W. The formula for the area of a rectangle is A = L * W, and we are given that the area must be at least 10,000 square feet. Substituting L = 280 - W into the area formula, we have (280 - W) * W >= 10,000. By solving this inequality, we can find the possible lengths of the parking lot.

Let's solve the inequality:

1. 280W - W^2 >= 10,000
2. -W^2 + 280W - 10,000 >= 0
3. W^2 - 280W + 10,000 <= 0

To find the possible values of W, we can use the quadratic formula: W = (-b ± sqrt(b^2 - 4ac)) / (2a).

Plugging in a = 1, b = -280, and c = 10,000, we get:

• W = (280 ± sqrt(280^2 - 4*1*10,000)) / (2*1)

After simplifying, we find two possible values for W: W = 10 and W = 270. Plugging these values back into the equation L = 280 - W, we find that the corresponding lengths are L = 270 and L = 10, respectively. Therefore, the possible lengths of the rectangular parking lot are 10 feet and 270 feet.

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