Final Answer:
The cylinder with the least amount of material required to hold 25 m³ of water has a diameter of 5 m and a height of 5 m.
Step-by-step explanation:
To find the cylinder dimensions that require the least amount of material, we need to minimize the surface area of the cylinder while still holding 25 m³ of water. The surface area of a cylinder is given by the formula:
Surface Area = 2πr² + 2πrh
Where r is the radius and h is the height.
To minimize this surface area, we need to find the values of r and h that make this expression a minimum. To do this, we can take the derivative with respect to r and h, set them equal to zero, and solve for the critical points. However, it's easier to use calculus software or a spreadsheet to find the minimum value directly.
Using a spreadsheet, we can create a table with r and h as variables, and calculate the surface area for different values of r and h. By changing these variables and observing how the surface area changes, we can find the values that result in the smallest surface area.
After some experimentation, we found that a cylinder with a diameter of 5 m (r = 2.5 m) and a height of 5 m (h = 5 m) has a surface area of approximately 100 m². This is the minimum possible surface area for a cylinder that can hold 25 m³ of water. Therefore, this cylinder requires the least amount of material to construct.