Determine The Following Limits

Determine The Following Limitsa Li

Determine the following limits: (a) limx→3 (2x² - x + 7) / (2x - 3) (b) limx→π cos x / 2 - sin x (c) limx→π/2 x cos² x / (1 - sin x)

A rectangular garden is to be constructed using a rock wall as one side of the garden and wire fencing for the other three sides. Given 131 ft of wire fencing, determine the dimensions that would create a garden of maximum area. What is the maximum area?

Paper For Above instruction

The problem of optimizing the area of a garden using a fixed length of fencing is a classic application of calculus, specifically optimization techniques. When a landowner chooses dimensions that maximize the area of a rectangular garden with one side against a wall, the key is to formulate the problem mathematically, derive the necessary condition for a maximum, and solve accordingly. The solution involves setting up a cost function (area in this case) and applying derivatives to find the critical points.

Let the length of the side parallel to the rock wall be denoted as x, and the width (perpendicular to the wall) as y. Since the rock wall forms one side of the garden, fencing is needed only for the remaining three sides—two widths and one length—totaling 131 feet. According to the fencing constraint, we get:

2y + x = 131

Express y in terms of x:

y = (131 - x)/2

The area A of the garden is given by:

A = x y = x (131 - x)/2 = (131x - x²)/2

To find the maximum area, differentiate A with respect to x and set the derivative equal to zero:

A'(x) = (131 - 2x)/2

Set derivative to zero:

(131 - 2x)/2 = 0 → 131 - 2x = 0 → 2x = 131 → x = 65.5 ft

Calculate y using x = 65.5 ft:

y = (131 - 65.5)/2 = 65.5/2 = 32.75 ft

Therefore, the dimensions of the garden for maximum area are approximately 65.5 ft (length along the wall) and 32.75 ft (width). The maximum area is:

A = x y = 65.5 32.75 ≈ 2144.125 square feet.

Hence, the garden should be approximately 65.5 feet long along the wall and 32.75 feet wide to maximize the area, which will be about 2144.125 square feet.

References

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