I need to enclose a section that is 250 square ft with a fence that cost $1.50 per yard, what dimension would i use to minimize the cost?
You don't specify what shape your enclosure is to have.
The shape which has the most area for the least perimeter of course is the circle.
so πr^2 = 250
r = 8.9206
and the circumference
= 2πr = 2π(8.9206)
= 56.05 ft
= 18.68 ft
So you will have to buy 19 yards for a cost of $28.50
If you want a 4-sided shape, it will be a square
side^2 = 250
side = 15.811
perimeter = 4s = 63.25 feet
= 21.08 yards.
in this case you will need 22 yards, since 21 yds are not enough
for a cost of $33
To minimize the cost of enclosing a section with a fence, we need to determine the dimensions that result in the least amount of fence needed.
Let's assume the width of the section is 'x' feet. To find the length of the section, we need to divide the total area (250 square feet) by the width: Length = 250 ft² / x ft.
The perimeter (P) of the enclosure can be calculated using the formula: P = 2 * (Length + Width).
Substituting the values, we get: P = 2 * (250 / x + x).
The cost of the fence per yard is given as $1.50. Since there are 3 feet in a yard, the cost per foot is $1.50 / 3 = $0.50 per foot.
The total cost of the fence (C) can be calculated by multiplying the cost per foot by the perimeter: C = P * $0.50.
Now we need to express the cost in terms of 'x' and find the value of 'x' that minimizes the cost. So, C = 2 * (250 / x + x) * $0.50.
To minimize the cost, we can take the derivative of C with respect to 'x', set it equal to zero, and solve for 'x'.
dC/dx = -500/x² + 1 = 0.
-500/x² = -1.
500/x² = 1.
x² = 500.
x = √500.
x ≈ 22.4.
Therefore, to minimize the cost, the width (and length) of the section should be approximately 22.4 feet.