Final answer:
To find the dimensions of the box that can be built with the least material, we can set up an optimization problem and solve for the critical points. By finding the derivative of the surface area equation with respect to x and setting it equal to zero, we can find the value of x that minimizes the surface area. The dimensions of the box that can be built with the least material are 2 cm x 2 cm x 29.5 cm.
Step-by-step explanation:
To find the dimensions of the box that can be built with the least material, we need to optimize the surface area of the box. Since the box has an open top and a square base, we can represent its dimensions as follows:
Let x be the length of the base side, and h be the height of the box.
The volume of the box is given as 118 cubic centimeters, so we have the equation:
x^2 imes h = 118
To minimize the material used, we need to minimize the surface area. The surface area of the box is given by:
A = x^2 + 4xh
We can substitute the volume equation into the surface area equation to get:
A = x^2 + 4x imes rac{118}{x^2}
Now, we can find the derivative of A concerning x and set it equal to zero to find the critical points:
rac{dA}{dx} = 2x - rac{472}{x^3} = 0
Solving this equation, we find x = 2 cm. Substituting this value back into the volume equation, we can solve for h:
2^2 imes h = 118
h = rac{118}{4} = 29.5 cm
Therefore, the dimensions of the box that can be built with the least material are 2 cm x 2 cm x 29.5 cm.