Final answer:
To find the exact dimensions of the box with the smallest possible surface area, use the volume equation and surface area equation to express the dimensions in terms of one variable. Then, find the critical points of the surface area function and solve for the variable to find the exact dimensions.
Step-by-step explanation:
To find the exact dimensions of the box with the smallest possible surface area, we need to optimize the surface area function. Let's start by writing equations for the dimensions in terms of one variable. Since the base length, ℓ, is three times the base width, w, we have ℓ = 3w. Let's also assume the height is h. The volume of the box is given as 330 cm³, so we have the equation ℓwh = 330.
Now, let's express the surface area of the box in terms of one variable, w. The surface area can be calculated as 2(ℓw + ℓh + wh). Substituting the value of ℓ from the first equation into the surface area equation, we get A(w) = 2(3w² + 3wh + wh).
To find the smallest possible surface area, we need to minimize A(w). We can do this by finding the critical points of A(w) and determining the values of w for which A'(w) = 0. By solving the resulting equation, we can find the exact dimensions of the box with the smallest possible surface area.
Learn more about Optimizing surface area of a closed box