Final answer:
To find the dimensions of the container with the least cost, we need to optimize the surface area of the container. The dimensions of the container of least cost are Side of Base (in) = x and Height (in) = 243/x².
Step-by-step explanation:
To find the dimensions of the container with the least cost, we need to optimize the surface area of the container.
Let's assume the side length of the square base is x in.
The volume of the container is given as 243 in³, so the height of the container is 243/x² in.
The surface area of the top and bottom is 2x² in², and the surface area of the sides is 4x(243/x²)x in².
The total cost of the materials is given by the formula C = (2x² + 4x(243/x²))($1/in²) + 4x(243/x²)($3/in²).
To find the dimensions of the container with the least cost, we need to minimize the cost function C with respect to x. We can do this by taking the derivative of C with respect to x, setting it equal to zero, and solving for x.
By solving this equation, we can find the value of x that minimizes the cost. Once we have x, we can calculate the height of the container using the formula h = 243/x².
So, the dimensions of the container of least cost are Side of Base (in) = x and Height (in) = 243/x².