Final answer:
To minimize the total surface area of each can, we need to find the base radius and height that give us the smallest surface area while maintaining a volume of 16π cm³. We can use the formulas for surface area and volume of a cylinder to set up an equation and then find the dimensions that minimize the surface area. By taking the derivative and solving for the critical points, we find that the base radius should be 2 cm and the height should be 4 cm.
Step-by-step explanation:
To minimize the total surface area of each can, we need to find the dimensions (base radius and height) that give us the smallest surface area while still maintaining a volume of 16π cm³.
The surface area of a cylinder consists of two circular end caps and the lateral surface area. The formula for the surface area of a cylinder is: A = 2πr² + 2πrh.
Let's use this formula and set it equal to the smallest possible surface area:
2πr² + 2πrh = min
We also know that the volume of a cylinder is given by the formula V = πr²h. Since we need to maintain a volume of 16π cm³, we can substitute this value into the equation:
πr²h = 16π
Now we can solve the volume equation for h:
h = 16/r²
Substitute this value for h back into the equation for surface area:
A = 2πr² + 2πr(16/r²) = 2πr² + 32π/r
To minimize this function, we can take the derivative with respect to r and set it equal to zero:
dA/dr = 4πr - (32π/r²) = 0
Solving this equation, we get r = 2.
Substitute this value back into the equation for h:
h = 16/(2²) = 16/4 = 4. Thus, the base radius of the cans should be 2 cm and the height should be 4 cm to minimize the total surface area while still maintaining a volume of 16π cm³.