The fence around Wayne Manor (a rectangular plot of land) is going to be replaced. No fence will be required for the side lying along Gotham river. If the new wrought iron fence costs $12 per meter for the side parallel to the river and $4 per meter for the other two sides, find the dimensions of the maximum area that can be enclosed by the fence if Bruce Wayne (AKA Batman) cannot spend more than $3600.
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To find the dimensions of the maximum area that can be enclosed by the fence, we need to set up a mathematical model and solve it.
Let's denote the width of the rectangular plot as x (which is the side parallel to the river) and the length of the rectangular plot as y (which are the other two sides).
The cost of the fence for the side parallel to the river would be 12x dollars, and the cost for the other two sides would be 4y dollars each. The total cost of the fence would be the sum of all these costs:
Total Cost = 12x + 4y + 4y
However, we are given that the total cost cannot exceed $3600, so we can write the following equation:
12x + 4y + 4y ≤ 3600
Simplifying this inequality, we get:
12x + 8y ≤ 3600
Now, let's express the area of the rectangular plot in terms of x and y. The area is given by:
Area = x * y
To find the dimensions of the maximum area, we need to maximize the Area function while satisfying the cost constraint. We can solve this problem using a technique called optimization.
We need to combine the cost constraint inequality with the area function. We can express the area as a function of one variable using the cost constraint:
Area = x * (3600 - 12x - 8y) / 8
Now, we have an equation for the area in terms of x. To find the maximum area, we need to find the critical points of this function. We can do this by taking the derivative of the area function with respect to x and setting it equal to zero:
d(Area)/dx = 0
Solving this equation will give us the value of x that corresponds to the maximum area.
Once we find the value of x, we can substitute it back into the cost constraint inequality to determine the corresponding value of y.
Finally, we can calculate the maximum area by plugging the values of x and y we found into the Area equation:
Area = x * y
This will give us the dimensions of the maximum area that can be enclosed by the fence.
If
x = side parallel to river
y = other sides
So, you want to maximize
a = xy
subject to
12x + 4*2y <= 3600
See what you can do with that.