rectangular area for cattle and uses a straight portion of a river as one side of the rectangle, as illustrated in the figure. Note that there is no fence along the river. If the farmer has 16001600 feet of fence, find the dimensions for the rectangular area that gives the maximum area for the cattle.
That's a lot of fencing..
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To find the dimensions for the rectangular area that gives the maximum area, we can use calculus. Let's assume the width of the rectangle is x, and the length is y.
Since there is no fence along the river, the perimeter of the rectangle can be calculated as follows:
Perimeter = 2x + y
We are given that the farmer has 1600 feet of fence, so we can write the equation:
2x + y = 1600
To find the maximum area, we need to express the area in terms of a single variable. The area of a rectangle is given by:
Area = xy
We can rewrite the equation we obtained earlier to solve for y:
y = 1600 - 2x
Substituting this into the area equation:
Area = x(1600 - 2x)
Now, we can maximize the area by finding the critical points. To do this, we take the derivative of the area equation with respect to x:
d(Area)/dx = 1600 - 4x
Setting this derivative equal to zero and solving for x, we find the value of x that gives us the maximum area:
1600 - 4x = 0
4x = 1600
x = 400
Now, we substitute the value of x back into one of the original equations to solve for y:
2x + y = 1600
2(400) + y = 1600
800 + y = 1600
y = 800
Therefore, the dimensions of the rectangular area that give the maximum area for the cattle are: width = 400 feet and length = 800 feet.