a rectangular dog pen is to be constructed so that one side is against an existing stone wall and the other three sides are to be fenced. of 500 feet of fence is to be used, determine the dimensions and area of the pen with maximum area.
I have no idea how to solve this problem.
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The max area is always a square.
You only need the fence for 3 sides, so divide 500/3
166.67 x 166.67 = max. area.
The maximum area occurs when the fencing is divided equally among lengths and widths. For all 4 sides fenced, that is a square. In this case, with the length parallel to the wall, we have length=500/2 = 250 and width = 500/2/2 = 125 so the area is 250*125 = 31,250
This is greater than 166.67^2 = 27,779.
To solve this algebraically,
If x is the length parallel to the wall and y is the width,
x+2y=500
The area is
a = xy = (500-2y)y = 500y-2y^2
da/dy = 500-4y
da/dy = 0 when y=125
so x = 500-2y = 250
To solve this problem, we need to find the dimensions of the rectangular dog pen that maximize its area using the given 500 feet of fence. Let's break down the problem into steps:
Step 1: Define the variables:
Let's assume the width of the pen as "w" (in feet) and the length of the pen as "l" (in feet). We need to find the values of "w" and "l" that maximize the area.
Step 2: Set up the equation for the perimeter:
Since we are given that the length of the pen is against an existing stone wall, there are three sides that need fencing. The equation for the perimeter is: w + l + w = 500.
Step 3: Simplify the equation:
From the equation in step 2, we can simplify it to 2w + l = 500.
Step 4: Rewrite the equation for length in terms of width:
We can rewrite the equation from step 3 to isolate the "l" variable. Subtracting 2w from both sides, we get l = 500 - 2w.
Step 5: Write the equation for area:
The area of a rectangle is calculated by multiplying the length and width. In this case, the area is given by A = w * l.
Step 6: Substitute the equation for l from step 4 into the area equation from step 5:
Substituting l = 500 - 2w into A = w * l, we get A = w * (500 - 2w).
Step 7: Simplify the area equation:
To find the maximum area, we can take the derivative of the area equation with respect to width, set it equal to zero, and solve for "w". However, to simplify things, we can first expand the equation derived in step 6: A = 500w - 2w^2.
Step 8: Find the derivative of the area equation:
Taking the derivative of A = 500w - 2w^2 with respect to "w", we get dA/dw = 500 - 4w.
Step 9: Set the derivative equal to zero and solve for "w":
Setting dA/dw = 500 - 4w equal to zero, we get 500 - 4w = 0. Solving for "w", we find w = 125.
Step 10: Substitute the value of "w" into the equation for length:
From step 4, we found l = 500 - 2w. Substituting w = 125 into this equation, we find l = 250.
Step 11: Calculate the area:
Using the dimensions obtained in steps 9 and 10, we can find the area of the pen by multiplying the width and length: A = w * l = 125 * 250 = 31250 square feet.
Therefore, the dimensions of the rectangular dog pen with the maximum area that can be constructed using 500 feet of fence are 125 feet by 250 feet, and the maximum area is 31,250 square feet.