An open box is to be constructed so that the length of the base is 4 times larger than the width of the base. If the cost to construct the base is 5 dollars per square foot and the cost to construct the four sides is 3 dollars per square foot, determine the dimensions for a box to have volume = 25 cubic feet which would minimize the cost of construction.
height?
dimensions of the base?
didn't I respond to this yesterday
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To determine the dimensions of the box that would minimize the cost of construction while having a volume of 25 cubic feet, we need to find the values for the height and the dimensions of the base.
Let's start by defining the variables:
Let width of the base = x feet
Then the length of the base = 4x feet
The height of the box will be calculated using the volume formula:
Volume = Length x Width x Height
25 = (4x) x x x Height
25 = 4x^2 x Height
Now, we can solve the equation for the height:
Height = 25 / (4x^2)
To minimize the cost of construction, we need to consider the cost of constructing the base and the sides separately.
The cost of constructing the base is given as 5 dollars per square foot. Since the base is a rectangle, the total cost of constructing the base can be calculated as:
Cost of Base = Length x Width x Cost per square foot
Cost of Base = (4x) x x x 5
Cost of Base = 20x^2 x 5
The cost of constructing the four sides is given as 3 dollars per square foot. Since the sides are all rectangles, the total cost of constructing the sides can be calculated as:
Cost of Sides = 2 x Height x (Length + Width) x Cost per square foot
Cost of Sides = 2 x Height x (4x + x) x 3
Cost of Sides = 2 x Height x 5x x 3
Cost of Sides = 30x^2 x Height
The total cost of construction is the sum of the cost of the base and the cost of the sides:
Total Cost = Cost of Base + Cost of Sides
Total Cost = 20x^2 x 5 + 30x^2 x Height
Now, substitute the value of the height in terms of x:
Total Cost = 20x^2 x 5 + 30x^2 x (25 / (4x^2))
Total Cost = 100x^2 + 750
To find the value of x that minimizes the total cost, we can take the derivative of the total cost with respect to x and set it equal to zero (in order to find the critical point).
d(Total Cost) / dx = 200x - 1500 = 0
200x = 1500
x = 1500 / 200
x = 7.5 feet
Therefore, the width of the base is 7.5 feet and the length of the base is 4 times larger, so the length is 4 * 7.5 = 30 feet.
To find the height, substitute the value of x into the height equation:
Height = 25 / (4x^2)
Height = 25 / (4 * 7.5^2)
Height = 25 / (4 * 56.25)
Height ≈ 0.111 feet (rounded to three decimal places)
Therefore, the height of the box is approximately 0.111 feet.