A rectangular box with square base (which is NOT necessarily a cube) and NO top is to be made to contain 9 cubic feet. The material for the base costs $2 per square foot and the material for the sides $3 per square foot. Find the dimensions that minimize the cost of the box AND �find the minimum cost of the box.
let the base be x by x ft
let the height be h ft
V = x^2 h
h = 9/x^2
Cost = 2(x^2) + 3(4xh)
= 2x^2 + 12x(9/x^2)
= 2x^2 + 108/x
dCost/dx = 4x - 108/x^2
= 0 for a min of Cost
4x = 108/x^2
x^3 = 27
x = 3
then h = 9/3^2 = 1
The min cost is when the base is 3 ft by 3 ft and the height is 1 foot
the minimum cost is 2x^2 + 108/x
= 18 + 108/9 = $30
Thanks for the help Reiny. Was wondering if you know how to setup the question bellow. Coz this is the one i find the hardest to do and had tried but don't get it?
A truck driver, on assignment from the owner of the truck is to drive on a 300 mile stretch of highway at a constant speed of v miles per hour. According to road signs, the minimum speed allowed is 55 miles per hour and the speed limit is 70 miles per hour. The cost of gas on the day of the trip is $2.60 per gallon and the gas in the truck has been measured to consumed at a rate of (1+(1/400)*(v^2) gallons per hour.
If the truck driver earns $20 per hour what speed v should the truck driver be assigned to drive in order to keep the cost for the owner of the company as low as possible? Find the minimum cost and compare it to the cost when the driver drives at 55 and 70 miles per hour.
To find the dimensions that minimize the cost of the box, we need to determine the length, width, and height of the box. Let's denote the length and width of the base as x, and the height as h.
Given that the box has a square base, we have x^2 as the area of the base.
The volume of the box is given as 9 cubic feet, so we have the equation:
x^2 * h = 9
To minimize the cost, we need to minimize the total cost, which is the sum of the cost of the base and the cost of the sides of the box.
The cost of the base is given as $2 per square foot, so the cost of the base is:
2 * x^2
The cost of the sides is given as $3 per square foot, and we need to determine the total area of the sides. The sides consist of the four vertical sides and the bottom (which is the base). So the total area of the sides is:
4 * x * h
The total cost is the sum of the cost of the base and the cost of the sides:
Total Cost = 2 * x^2 + 4 * x * h
To minimize the cost, we need to minimize the Total Cost function.
To find the minimum cost, we can differentiate the Total Cost function with respect to x and set it equal to zero. Let's find ∂(Total Cost)/∂x:
∂(Total Cost)/∂x = 4x + 4h * (∂h/∂x)
Since h = 9 / (x^2), we can find ∂h/∂x using the quotient rule:
∂h/∂x = -18x / (x^2)^2
Simplifying, we have:
∂h/∂x = -18x / x^4 = -18 / x^3
Now we can substitute this into ∂(Total Cost)/∂x:
0 = 4x + 4h * (∂h/∂x)
0 = 4x + 4 * (9 / (x^2)) * (-18 / x^3)
0 = 4x - 4 * (9 * 18) / (x^5)
0 = 4x - 4 * (162) / (x^5)
0 = 4x - 648 / (x^5)
0 = 4x^6 - 648
Solving for x, we have:
4x^6 = 648
x^6 = 162
x = (162)^(1/6)
x ≈ 2.386
Now we can find the height h using the equation:
h = 9 / (x^2)
h = 9 / (2.386^2)
h ≈ 1.857
Therefore, the dimensions that minimize the cost of the box are approximately:
Length = Width = x ≈ 2.386
Height = h ≈ 1.857
To find the minimum cost of the box, we substitute the values of x and h into the Total Cost function:
Total Cost = 2 * x^2 + 4 * x * h
Total Cost = 2 * (2.386)^2 + 4 * 2.386 * 1.857
Total Cost ≈ $23.53
Therefore, the minimum cost of the box is approximately $23.53.
To find the dimensions that minimize the cost of the box, we need to consider the cost function. Let's assume the base has side length x and the height of the box is h.
The volume of the box is given as 9 cubic feet, so we have x^2 * h = 9.
Let's denote the cost function as C(x, h) which is the cost of constructing the box. The cost is calculated by summing the cost of the base and the cost of the four sides.
The cost of the base is given as $2 per square foot, so it is 2 * x^2.
The cost of the four sides is given as $3 per square foot, and the total surface area of the four sides is 2xh. Therefore, the cost of the sides is 3 * 2xh = 6xh.
Now we can rewrite the cost function as C(x, h) = 2x^2 + 6xh.
To find the dimensions that minimize the cost, we need to take partial derivatives of the cost function with respect to x and h and set them equal to zero.
∂C/∂x = 4x + 6h = 0 (equation 1)
∂C/∂h = 6x = 0 (equation 2)
From equation 2, we can see that x = 0 or h = 0 is a solution. However, since we cannot have a box with zero dimensions, we can ignore those solutions.
From equation 1, we can solve for h in terms of x:
6h = -4x
h = -4x/6
h = -2x/3
Now substitute this expression for h back into the equation x^2 * h = 9:
x^2 * (-2x/3) = 9
-2x^3/3 = 9
-2x^3 = 27
x^3 = -27/-2
x^3 = 27/2
x = (27/2)^(1/3)
x ≈ 2.996
We can substitute this value of x back into the expression h = -2x/3:
h = -2(2.996)/3
h ≈ -1.997
Since we can't have a negative height for the box, we need to discard this solution.
Now we have the approximate dimensions of the box: x ≈ 2.996, h ≈ -1.997.
To find the minimum cost, substitute these values into the cost function:
C(x, h) = 2x^2 + 6xh
C ≈ 2(2.996)^2 + 6(2.996)(-1.997)
C ≈ 17.98
Therefore, the dimensions that minimize the cost of the box are x ≈ 2.996 and h ≈ -1.997, and the minimum cost is approximately $17.98.