Final answer:
To find the dimensions of the rectangular box with the largest possible volume among all rectangular boxes with a fixed surface area of 20 square meters, we need to consider the relationship between surface area and volume. By solving the equations and calculating the volume for each option given, we can determine that the box with the dimensions 5m×5m×2m has the largest possible volume.
Step-by-step explanation:
The question asks us to find the dimensions of the rectangular box with the largest possible volume among all rectangular boxes with a fixed surface area of 20 square meters. To find the dimensions with the largest volume, we need to consider the relationship between surface area and volume of a rectangular box. The formula for the surface area of a rectangular box is:
Surface Area = 2lw + 2lh + 2wh
Where l, w, and h are the length, width, and height of the box, respectively. Given that the surface area is fixed at 20 square meters, we have:
2lw + 2lh + 2wh = 20
Now, we need to express the volume of the box in terms of one variable. The formula for the volume of a rectangular box is:
Volume = lwh
We can rewrite the surface area equation as:
lw + lh + wh = 10
Solving this equation for one variable, we can obtain:
h = (10 - lw)/(l + w)
Substituting this expression for h in the volume equation, we get:
Volume = lwh = l(10 - lw)/(l + w)
To find the dimensions of the box with the largest possible volume, we can either solve this equation analytically or use optimization techniques. After calculating the volume for each option given, we can determine that the box with the dimensions a) 5m×5m×2m has the largest possible volume.