SOLUTION: A farmer has enough money to build only 100 meters of fence. What are the dimensions of the field he can enclose with the maximum area?

Question 535681: A farmer has enough money to build only 100 meters of fence. What are the dimensions of the field he can enclose with the maximum area?
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Be careful. This could be a "trick" question. If it is required that the field be a rectangular shape, the maximum area would be enclosed by a field that is square. In that case the field would have all four sides be of length 25 meters. This would give you a perimeter of 100 meters and would enclose a field of dimensions 25 meters by 25 meters, resulting in an enclosed area of 25 times 25 = 625 square meters.
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But if the farmer is allowed to build a circular fenced area, this will result in a different area being enclosed. The 100 meters of fence would be the circumference of the field. Since we know that the formula for circumference of a circle is:
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Where C represents the circumference (in this case C = 100 meters) and R is the radius of the circular field. We can solve for the radius of this field by dividing both sides of the equation by to get:
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And substituting 100 for C the equation becomes:
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Dividing the numerator of 100 by results in finding that the radius R is 15.91549431 meters.
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We also know from geometry that the Area of a circle is given by the formula:
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in which A represents the area and R again represents the radius. We know that for this problem R equals 15.91549431 meters. Substitute this value into the equation for the area and you get:
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Square the radius and multiply that result by and you find that the circular field contains an area of 795.7747154 square meters.
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This is more area than if the field were a square of 25 meters by 25 meters. The area made by fencing a circle from 100 meters of fence contains nearly 171 square meters more (795.7747154 square meters minus 625 square meters = 170.7747154 square meters).
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Hope this helps you understand the problem a little better.
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