Question

If f' is continuous, f(4)=0 and a box with an open top has vertical sides, a square bottom, and a volume of 108 cubic meters, if the box has the least possible surface area, find its dimensions. If f'(4)=10, evaluate lim x → 0?

Answer 1

To find the dimensions of the box with the least possible surface area, take the derivative of the surface area function and set it equal to zero. Use the equation for the volume of the box to solve for one variable in terms of the other. The given information regarding f'(4) does not help evaluate the limit lim x → 0.

Step-by-step explanation:

To find the dimensions of the box with the least possible surface area, we need to minimize the surface area function by taking its derivative and setting it equal to zero. Let's denote the side length of the square bottom as x and the height of the box as h. Then, the surface area function, S(x, h), is given by:

S(x, h) = x^2 + 4xh

We know that the volume of the box is 108 cubic meters, so we have the equation:

x^2h = 108

We also know that f'(4) = 10. However, the limit lim x → 0 is unrelated to the given information, so I will not be able to evaluate it.