A fence is to be built to enclose a rectangular area of 310 square feet. The fence along three sides is to be made of material that costs 3 dollars per foot, and the material for the fourth side costs 14 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.
Let two of the sides be x ft each
let the other two sides by y ft each
xy = 310
y = 310/x
cost = 3(2x) + 3y + 14y
= 6x + 17y
= 6x + 17(310/x)
= 6x + 5270/x
d(cost)/dx = 6 - 5270/x^2
=0 for a min of cost
6 = 5270/x^2
6x^2 = 5270
x^2 = 878.333..
x = √878.333
= appr 29.64 ft
y = 310/29.64 = 10.46
To find the dimensions of the enclosure that is most economical to construct, we need to minimize the cost of the fence.
Let's assume the length of the rectangle is 'x' feet, and the width is 'y' feet. The formula for the area of a rectangle is A = length * width. In this case, we have the area, which is 310 square feet, so we can set up the equation:
xy = 310
Since we want to minimize the cost, we need to find the cost function.
The total cost of the fence is given by:
C = 3(2x + y) + 14x
The first term, 3(2x + y), represents the cost of the three sides, where the material costs $3 per foot. The second term, 14x, represents the cost of the fourth side, where the material costs $14 per foot.
Now, we need to express one variable in terms of the other so that we can substitute it into the cost function. We can solve the area equation for y:
y = 310 / x
Substituting this into the cost function, we get:
C = 3(2x + 310 / x) + 14x
Simplifying further:
C = 6x + 930 / x + 14x
Now, we have the cost function C in terms of x only. To minimize the cost, we can take the derivative of C with respect to x and set it equal to zero:
dC/dx = 6 - 930 / x^2 + 14 = 0
To solve this, we can multiply through by x^2:
6x^2 - 930 +14x^3 = 0
Simplifying further:
14x^3 + 6x^2 - 930 = 0
We can use numerical methods to solve for x, but in this case, we can see that x = 5 is a root of this equation. Hence, one side of the rectangle is 5 ft.
Substituting this value back into the area equation, we can find y:
5y = 310
y = 310 / 5
y = 62
So, the dimensions of the enclosure that is most economical to construct are 5 feet by 62 feet.
To find the dimensions that are most economical to construct, we need to minimize the cost of the fence.
Let's assume the length of the enclosure is x feet and the width is y feet.
The area of the enclosure is given as 310 square feet, so we have the equation:
x * y = 310
The cost of the three sides (the length and two widths) is 3 dollars per foot, so the cost of these three sides is:
C1 = 3 * (x + 2y)
The cost of the fourth side (the other width) is 14 dollars per foot, so the cost of this side is:
C2 = 14 * x
The total cost of the fence is the sum of these two costs:
C = C1 + C2 = 3 * (x + 2y) + 14 * x
To minimize the cost, we can differentiate the cost function C with respect to x and y, and set the derivatives equal to zero:
dC/dx = 3 + 14 = 0
dC/dy = 6 = 0
Solving these equations, we find x = 26 and y = 26 as the dimensions that minimize the cost of the fence.
Therefore, the most economical dimensions of the enclosure are a length of 26 feet and a width of 26 feet.