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I am taking a calculus class and i need help with the following problem:
An athletic field consists of a rectangular region with a semicircular region at each end. The perimeter will be used for a 440 yard track. Find the value of x for which the area of the rectangular region is as large as possible. I JUST NEED TO KNOW HOW TO GET STARTED AND WHAT FORMULA I NEED TO USE. If anyone could please help
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what does x refer to? the radius of the semi circular region, edge of rectangular with semi circle? or what?
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x=????????
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I believe x may refer to a side of the rectangle.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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I think refers to the side of the rectangle and this was a optimiaztion problem were you use a derivitive formula to help you with finding the answer. Anyone know about derivtive formalas?
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But which side of the rectangle is x? I'm pretty sure it's not going to matter though.
Perimeter = 440 = 2y + 2*pi*r
2d = r
x = d
2x = r
r = 1/2x
perimeter = 440 = 2y + pi*x
2y = 440 - pi*x
y = 220 - pi*x/2
area of the rectangle = x * y = x (220 - pi*x/2)
So
we need to find the value for x for which f(x) = x(220 - pi*x/2) is at it's max.
Here's where the calculus comes in... How do we find the max?
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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thanks so much very helpful
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