Final answer:
To maximize the volume of a rectangular box with a square base using a 100 cm wire, set up an equation to find the best length 'x' for the sides of the base to calculate the height 'h' and then the volume 'V = x²h'. Differentiate the volume function to find the maximum volume.
Step-by-step explanation:
Maximizing the Volume of a Rectangular Box with a Square Base
To answer this question, we need to apply mathematical concepts to optimize the volume of the box. With a wire that is 100 cm long, we can calculate the largest possible volume of a box with a square base that can be formed by cutting the wire and using it as the skeleton of the box. Since a square base requires four equal sides and the rectangular box will also have four vertical sides and four sides on the top, let's denote each side of the square base as 'x'. Thus, the total used wire length would be '12x' because the box needs 4 'x' lengths for the base, 4 'x' lengths for the top, and 4 'x' lengths for the vertical sides.
To maximize the volume 'V' of the box, which is given by the formula 'V = x²h' where 'h' is the height, we need first to express the height in terms of 'x'. Since we have 100 cm of wire, the equation for total length is:
12x = 100
Therefore, we can express 'h' as (100 cm - 8x) / 4. Finally, we have the volume function V(x) = x² * (100 - 8x) / 4 which we can differentiate and find the maximum volume by setting the first derivative equal to zero and solving for 'x'. The value of 'x' that maximizes the volume can then be substituted back into the volume equation to find the largest possible volume.