A rectangular storage container with an open top is to have a volume of k cubic meters. The length of its base is twice its width. The material for the base costs $6 per square meters and the material for the sides costs $10 per square meter.
y=2x
xyz=k, so z=k/2x^2
the cost c is
c(x,y,z) = 6xy + 10(2xz + 2yz)
c(x) = 6x(2x) + 20(3x)(k/2x^2)
= 12x^2 + 30k/x
To find the dimensions of the rectangular storage container, you can start by assigning variables. Let's say that the width of the base is x meters.
Since the length of the base is twice the width, the length would be 2x meters.
The height of the container which corresponds to the side of the rectangular container can be represented by h meters.
The volume of a rectangular container is given by the formula V = length * width * height. In this case, the volume is k cubic meters, so we have:
k = (2x) * x * h
k = 2x^2 * h
Now, let's calculate the surface area of the base and the four sides of the container.
The surface area of the base, which is a rectangle, is given by the formula A_base = length * width. In this case, it would be:
A_base = (2x) * x
A_base = 2x^2 square meters
The surface area of each side of the container is equal to the height multiplied by the sum of the length and width. Since there are four sides, the total surface area of the sides would be:
A_sides = 4 * ((2x * h) + (x * h))
A_sides = 4 * (3xh)
A_sides = 12xh square meters
The total surface area of the container is the sum of the surface area of the base and the surface area of the sides. So, the total surface area would be:
A_total = A_base + A_sides
A_total = 2x^2 + 12xh square meters
Now, we can calculate the cost of the materials. The cost of the material for the base is $6 per square meter and the cost of the material for the sides is $10 per square meter.
The cost of the material for the base would be:
Cost_base = A_base * $6
Cost_base = (2x^2) * $6
Cost_base = 12x^2 dollars
The cost of the material for the sides would be:
Cost_sides = A_sides * $10
Cost_sides = (12xh) * $10
Cost_sides = 120xh dollars
The total cost of the materials would be the sum of the cost of the base and the cost of the sides. So, the total cost would be:
Total_cost = Cost_base + Cost_sides
Total_cost = 12x^2 + 120xh dollars
Now, depending on the specific value of k, you can solve for either x or h. If you have a specific value of k, you can substitute it into the equation for volume:
k = 2x^2 * h
Then, solve for either x or h in terms of the other variable. Once you know the value of one variable, you can substitute it back into the total cost equation and find the total cost of the materials.