Final answer:
To minimize the cost of manufacturing the rectangular box with a square base and a given volume, we need to find the dimensions that result in the minimum surface area. Using the volume formula and the cost equations for the base and sides, we can express the cost in terms of 'x' and 'h'. By differentiating the cost equation, we can find the value of 'x' that minimizes the cost. Substituting this value back into the equation for 'h', we can determine the dimensions and the minimum cost.
Step-by-step explanation:
To minimize the cost of manufacturing the rectangular box, we need to find the dimensions that will result in the minimum surface area. Let's assume the side length of the square base is 'x' cm. The height of the box, which is also the length of the sides, will be 'h' cm.
The volume of the box is given as 390 cm³, so we have:
Volume = x²h = 390
From this equation, we can express 'h' in terms of 'x': h = 390/x²
The surface area of the base is x², and the surface area of the sides is 4xh. The total cost of manufacturing is:
Cost = (0.10 * x²) + (0.20 * 4xh)
Substituting the expression for 'h' into the cost equation, we get:
Cost = (0.10 * x²) + (0.20 * 4x * (390/x²))
To find the minimum cost, we need to differentiate the cost equation with respect to 'x', set it equal to zero, and solve for 'x'.
After finding the value of 'x', we can substitute it back into the equation for 'h' to find its corresponding value. Finally, we can calculate the minimum cost by substituting 'x' and 'h' into the cost equation.