A farmer has 100 feet of fencing and wants to enclose a rectangular field. What is the maximum area of the field that he can enclose?
To find the maximum area of the rectangular field, we can use the formula:
Area = Length x Width
Since the farmer has 100 feet of fencing, the perimeter of the field is 100 feet. The perimeter of a rectangle is given by:
Perimeter = 2(Length + Width)
Therefore, we have:
100 = 2(Length + Width)
Solving for Length, we get:
Length = 50 - Width
Substituting this expression for Length into the formula for Area, we get:
Area = (50 - Width) x Width
To find the maximum area, we need to find the value of Width that maximizes this expression. We can do this by taking the derivative of Area with respect to Width and setting it equal to zero:
d(Area)/d(Width) = 50 - 2Width = 0
Solving for Width, we get:
Width = 25 feet
Substituting this value back into the expression for Length, we get:
Length = 25 feet
Therefore, the maximum area of the field that the farmer can enclose is:
Area = Length x Width = 25 feet x 25 feet = 625 square feet
Related Questions:
- What is the perimeter of the field? 100 feet
- What is the length of the field? 25 feet
- What is the width of the field? 25 feet
- What is the formula for the area of a rectangle? Area = Length x Width
- What is the formula for the perimeter of a rectangle? Perimeter = 2(Length + Width)
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