A rectangular storage container with an open top is to have a volume of 15m^3. The length of the base is twice the width of the base. Material for the base costs $6 per square meter. Material for the sides costs $7 per square meter. Find the cost of the materials for the cheapest such container. Round answer to nearest hundredth.
I know that you need to find an equation, get the derivative, find the critical numbers, and then find the minimum. But, I have no clue on how to find the equation! If someone could please help me!
You will need a diagram.
Visualize the box "flattened", you have a rectangular base, with rectangles on each of its sides.
These four rectangles must have the same width, namely the height of the box.
Let that height be h m.
Let the width of the base be x m, then its length is 2x m.
So the volume is x(2x)h or 2hx^2 m^3
but we know this is 15.
So one of the equations is 2hx^2 = 15 or
h = 7.5/x^2
Since you want to minimize the Cost, you now need an equation for Cost
Cost of base = 6(x)(2x)=12x^2
Cost of sides = 2(7)(x)h) + 2(7)(2x)(h)
=42xh
Cost = 12x^2 + 42xh
=12x^2 + 42x(7.5/x^2)
=12x^2 + 315/x
Cost' = 24x - 315/x^2 = 0 for a min of Cost
Solve this....i got x=2.3588....
You actually have to plug it into the original cost equation and I get $200.31 and it's the right answer! Thank you so very much for your help!!!!
To find the equation for the cost of materials, start by determining the dimensions of the storage container. Based on the given information, let's assume the width of the base is x meters. Since the length of the base is twice the width, the length would be 2x meters.
Next, determine the height of the container. Since the volume of the container is 15m^3, you can set up an equation using the formula for the volume of a rectangular prism:
Volume = length * width * height
15 = (2x) * x * h
Simplify this equation to find h in terms of x:
15 = 2x^2 * h
h = 15 / (2x^2)
h = 7.5 / x^2
Now that we have the dimensions of the container, we can calculate the cost. The cost of the base of the container can be found by multiplying the area by the cost per square meter, which is $6:
Cost of base = 6 * (2x) * x = 12x^2
The cost of the four sides of the container can be found by multiplying the area by the cost per square meter, which is $7. Since we have four identical sides, we multiply by 2:
Cost of sides = 2 * 7 * (x * h) + 2 * 7 * (2x * h)
= 14xh + 28xh
= 42xh
The total cost of materials is the sum of the cost of the base and the cost of the sides:
Cost = Cost of base + Cost of sides
= 12x^2 + 42xh
= 12x^2 + 42x(7.5 / x^2)
= 12x^2 + 315 / x
To find the minimum cost, take the derivative of the cost equation with respect to x:
Cost' = 24x - 315 / x^2
Set Cost' equal to zero and solve for x:
24x - 315 / x^2 = 0
Solve this equation to find the x-coordinate of the minimum cost. In this case, you found x ≈ 2.3588.
Finally, substitute this value of x back into the original cost equation to find the minimum cost:
Cost = 12x^2 + 315 / x
= 12(2.3588)^2 + 315 / 2.3588
≈ $200.31
Therefore, the cost of materials for the cheapest container is approximately $200.31.