Final Answer:
The dimensions that minimize the surface area of the airtight fuel tank with a specified volume V m³ consist of a right circular cylinder with a hemisphere at one end.
Step-by-step explanation:
The problem involves optimizing the surface area of an airtight fuel tank with a given volume. The tank's geometry is described as a combination of a right circular cylinder and a hemisphere. To minimize the surface area, the critical step is to express the surface area in terms of a single variable.
Let's denote the radius of the cylinder as 'r' and the common radius of the hemisphere and cylinder as 'R'. The volume of the tank, V, can be expressed as the sum of the volumes of the cylinder and hemisphere.
To minimize surface area, we need to find the critical points of the surface area function by taking the derivative with respect to the variable 'r'. The result will lead to an equation that relates 'r' and 'R'. Solving this equation will provide the values of 'r' and 'R' that minimize the surface area while maintaining the specified volume.
This optimization problem aligns with real-world engineering challenges, where minimizing materials while meeting specific criteria is crucial. It also showcases the application of calculus in solving practical problems related to geometry and optimization.