A fisherman has 2300 feet of net and wants to cover a rectangular pond facing a building on
the longer side of the pond. He needs no net along the building side. What are the dimensions of
the pond that will give the largest area
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A fisherman has 2300 feet of net and wants to cover a rectangular pond facing a building on
the longer side of the pond. He needs no net along the building side. What are the dimensions of
the pond that will give the largest area
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Answer:
the dimensions of the rectangular pond that will give the largest area are:
Length (L) = 1150 feet
Width (W) = 575 feet
Step-by-step explanation:
Given:
Total amount of net (perimeter excluding the side along the building) = 2300 feet.
Let the length of the rectangular pond be L (in feet) and the width (perpendicular to the building) be W (in feet).
1. Write the constraint equation based on the total amount of net:
L + 2W = 2300
2. Express one variable in terms of the other using the constraint equation:
L = 2300 - 2W
3. Write the area equation for the rectangular pond:
Area (A) = Length (L) * Width (W) = L * W
4. Substitute the expression for L from the constraint equation into the area equation:
A = (2300 - 2W) * W
5. Simplify the area equation:
A = 2300W - 2W^2
6. The goal is to find the value of W that maximizes the area (A). The vertex formula for a quadratic function in the form f(x) = ax^2 + bx + c is given by x = -b / (2a).
7. In our case, the quadratic function is A = -2W^2 + 2300W. Comparing with the quadratic form, we have a = -2 and b = 2300.
8. Find the value of W that gives the maximum area:
W = -2300 / (2 * -2) = 575
9. Use the constraint equation to find the corresponding value of L:
L = 2300 - 2 * 575 = 1150
So, the dimensions of the rectangular pond that will give the largest area are:
Length (L) = 1150 feet
Width (W) = 575 feet.