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Chapter 2: Problem 76

A rectangular parking lot with a perimeter of 440 feet is to have an area of at least 8000 square feet. Within what bounds must the length of the rectangle lie?

Short Answer

Expert verified
The length of the rectangle must lie within the bounds of approximately 79.1 feet and 140.9 feet.

Step by step solution

01

Step 1:Derive the relationships

We've two equations based on the problem constraints. From the perimeter, we can derive the equation \(2L + 2B = 440\) and solve for B to have \( B = 220 - L\). As for the area, we get \(L*B = 8000\) or \(L*(220 - L) = 8000\)
02

Find the minimum and maximum length

We need to find the values for L that satisfy the area condition. We need to obtain the roots of the quadratic equation \(L ^2 - 220L + 8000 = 0\). Using the quadratic formula \(L = \frac{-(-220) 卤 \sqrt{(-220)^2 - 4*1*8000}}{2*1}\), we get L can be either approximately 140.9 or 79.1.
03

Verify the solutions

Because the length L is greater than or equal to the breadth B, only the larger value, approximately 140.9, is a plausible solution. Hence the length must be less than or equal to 140.9 feet.

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