a company is designing shipping crated and wants the volume of each crate to be 4 cubic ft and the crate;s base to be a square between 1 and 1.5 ft per side. if the material for the bottom costs $5, and the side $3 and the top $1 per square foot. what dimension will give the minimum cost?
let the base be x by x
and let the height be y
volume = x^2 y
4 = x^2y
y = 4/x^2
cost = 5(x^2) + 4(3)xy + 1x^2
= 6x^2 + 12xy
= 6x^2 + 12x(4/x^2)
= 6x^2 + 48/x
d(cost)/dx = 12x - 48/x^2 = 0 for a min of cost
12x = 48/x^2
x^3 = 4
x = 4^(1/3) = appr. 1.587
which is a bit more than the 1.5 allowed
so we have to stay with 1.5 for each side of the base
then y = 4/1.5^2 = 1.78
base = 1.5 by 1.5, height = 1.78
Well, buckle up for this one! Let's do some math and find the most cost-effective dimensions for the shipping crate.
First, let's express the dimensions of the square base as "x" since all four sides will be equal. The height of the crate can be expressed as "h".
The volume of a crate is given by the formula V = length × width × height. Since the base is a square with sides of length "x", the volume formula for this crate becomes V = x²h.
We want the volume of each crate to be 4 cubic ft, so we have the equation x²h = 4.
Now, let's consider the cost of each part of the crate:
- The bottom, which has an area of x² square feet, costs $5 per square foot.
- All four sides combined have an area of 4xh square feet, costing $3 per square foot.
- The top, which is also x² square feet, costs $1 per square foot.
So, the total cost (C) is given by the equation C = 5x² + 3(4xh) + x².
Now, we can simplify this equation to find the minimum cost:
C = 5x² + 12xh + x²
= 6x² + 12xh
To minimize cost, we need to find the values of x and h that satisfy the constraints and minimize the equation above.
Since the crate's base must be a square between 1 and 1.5 ft per side, we know that 1 ≤ x ≤ 1.5.
To solve the problem and find the minimum cost, we need more information about the height (h) constraints. Are there any constraints for h?
Once we have all the necessary information, we can solve the problem and find the dimensions that give the minimum cost for the crate. Feel free to provide the additional information, and I'll be here to assist you with the calculations!
To find the dimensions that will give the minimum cost, we need to consider the cost function for making the crate. Let's assume the side of the square base is "x" feet.
Volume of the crate = 4 cubic ft
Since the base is a square, the area of the base = x^2 square ft
Height of the crate = Volume / Base area = 4 / x^2 ft
Cost of the bottom = Area of the bottom * Cost per square foot = x^2 * $5 = 5x^2 dollars
Cost of the sides (including all four sides) = 4 * Height * Length * Cost per square foot
= 4 * (4 / x^2) * x * $3 = 48 / x dollars
Cost of the top = Area of the top * Cost per square foot = x^2 * $1 = x^2 dollars
Total cost = Cost of bottom + Cost of sides + Cost of top
= 5x^2 + 48/x + x^2
Now, we can find the derivative of the cost function with respect to x to find the critical points:
d(Cost)/dx = d(5x^2 + 48/x + x^2)/dx
= 10x - 48/x^2 + 2x
Setting the derivative equal to zero to find the critical points:
10x - 48/x^2 + 2x = 0
12x - 48/x^2 = 0
12x^3 - 48 = 0
x^3 = 4
x = ∛4
x ≈ 1.5874 ft
Since the base length should be between 1 and 1.5 ft per side, we can round x to 1.5 ft.
Therefore, the base dimension that will give the minimum cost is a square with 1.5 ft sides.
To find the dimensions that will give the minimum cost, we need to determine the dimensions of the crate that will minimize the cost function.
Let's assume that the side length of the base of the crate is 'x' feet. Since the base is a square, the length and width of the base will both be 'x'.
The height of the crate can be calculated by dividing the volume (4 cubic ft) by the area of the base. The volume of a rectangular prism (in this case, the crate) is given by the formula: Volume = Length × Width × Height. In this case, the length and width are 'x', and the volume is 4 cubic ft. So, the equation is: 4 = x * x * Height.
Simplifying this equation, we have:
4 = x^2 * Height
We can rewrite this equation as:
Height = 4 / x^2
Now, let's calculate the surface area and total cost using the given cost values.
Surface Area = Top Area + Bottom Area + 4 × Side Area
Top Area = Length × Width = x * x = x^2
Bottom Area = x * x = x^2
Side Area = Length × Height = x * (4 / x^2) = 4 / x
Surface Area = x^2 + x^2 + 4 / x = 2x^2 + 4 / x
Total Cost = (Surface Area of Bottom) × $5 + (Surface Area of Side) × $3 + (Surface Area of Top) × $1
Total Cost = x^2 * $5 + (4 / x) * $3 + x^2 * $1
Total Cost = 5x^2 + 12 / x + x^2
To find the value of 'x' that minimizes the cost, we need to differentiate the cost function with respect to 'x' and solve for x when the derivative equals zero. Let's do that:
d(Cost) / dx = 0
d(5x^2 + 12 / x + x^2) / dx = 0
10x - 12 / x^2 + 2x = 0
Multiplying through by x^2 to eliminate the fraction:
10x^3 - 12 + 2x^3 = 0
12x^3 + 10x^3 - 12 = 0
22x^3 - 12 = 0
22x^3 = 12
x^3 = 12 / 22
x^3 = 6 / 11
x = (6 / 11)^(1/3)
Thus, the dimension that will give the minimum cost is approximately:
x = 0.8828 ft (rounded to four decimal places)
Therefore, the side length of the base of the crate that will minimize the cost is approximately 0.8828 feet.