gardner wishes to encloe a rectangular 3000 square feet area with bushes on three sides and a fence on the 4th side .If the bushes cost $25.00per foot and the fence costs $10.00 per foot, find the dimensions that minimize the total cost and find the minimum cost
let the length be x feet, and the width be y feet
we know xy = 3000
so y = 3000/x
You don't say if the fence is along the length or along the width.
I will work it as if it the length has the fence.
If otherwise, just go through it again by changing the equation.
Cost = 10x + 10y + 25(2y)
= 10x + 60y
= 10x + 60(3000/x)
d(cost)/dx = 10- 180000/x^2 = 0 for a minimum Cost
10 = 180000/x^2
x^2 = 18000
x = 134.16
y = 3000/x = 22.36
minimum cost = 10(134.16) + 60(22.36) = 2683.28
thank u sooo much
To find the dimensions that minimize the total cost, we need to set up an equation for the cost in terms of the dimensions of the rectangular area.
Let's assume the length of the rectangle is L and the width of the rectangle is W.
Since the area of the rectangle is given as 3000 square feet, we have the equation:
L * W = 3000
Now, let's calculate the cost of the bushes and the fence.
The cost of the bushes on three sides will be the perimeter of the rectangle (excluding the side with the fence), multiplied by the cost per foot of bushes ($25.00):
Cost of bushes = 3 * (L + W) * $25.00 = 75(L + W)
The cost of the fence on the fourth side will be the length of that side (which is equal to L) multiplied by the cost per foot of the fence ($10.00):
Cost of fence = L * $10.00 = 10L
The total cost is the sum of the cost of bushes and the cost of the fence:
Total cost = Cost of bushes + Cost of fence = 75(L + W) + 10L
To minimize the total cost, we need to find the values of L and W that minimize this equation.
Next, we can use the given equation L * W = 3000 to express one variable in terms of the other and substitute it into the total cost equation.
Solve the area equation for W:
W = 3000 / L
Substitute this value of W into the total cost equation:
Total cost = 75(L + (3000 / L)) + 10L
Now, we have the total cost equation in terms of a single variable, L.
To find the minimum cost, we can differentiate this equation with respect to L to find its critical points (where the derivative is equal to zero), and then calculate the cost at those critical points.
Differentiating the total cost equation with respect to L gives us:
d(Cost)/dL = 75 - (2250000 / L^2) + 10
Setting this derivative equal to zero and solving for L will give us the value of L that minimizes the cost.
75 - (2250000 / L^2) + 10 = 0
2250000 / L^2 = 85
L^2 = 2250000 / 85
L^2 = 26470.58824
L ≈ √(26470.58824)
L ≈ 162.732
Substituting this value of L back into the area equation, we can find the corresponding width:
W = 3000 / L ≈ 3000 / 162.732 ≈ 18.422
So, the dimensions that minimize the total cost are approximately L ≈ 162.732 feet and W ≈ 18.422 feet.
To find the minimum cost, substitute these values of L and W back into the total cost equation:
Total cost = 75(L + W) + 10L
Total cost ≈ 75(162.732 + 18.422) + 10(162.732)
Total cost ≈ 13725.9 + 1627.32
Total cost ≈ $15,353.22
Therefore, the minimum cost to enclose a rectangular 3000 square feet area is approximately $15,353.22.