Final answer:
To find the minimum total surface area of the half cylinder, we need to consider the volume and surface area formulas. By taking the derivative of the surface area formula and solving for the radius, we find that the radius is equal to the volume cubed root divided by π. Substituting this value back into the surface area formula, we can simplify and find the ratio of the height to the diameter as π / (π+2π^(1/3)).
Step-by-step explanation:
To find the minimum total surface area, we need to consider the volume and surface area formulas for the given shape. The volume of the cylinder is given by V = πr²h, and the external surface area is A = 2πr² + 2πrh. We can rewrite the volume formula as h = V / (πr²), and substitute this into the surface area formula to get A = 2πr² + 2πr(V / (πr²)). Simplifying, we have A = 2πr² + 2V/r. To minimize A, we can take the derivative with respect to r and set it equal to zero. Solving this equation gives us r = (V/π)^(1/3), which means the radius is equal to the volume cubed root divided by π. Substituting this back into the surface area formula, we get A = 2π(V/π)^(2/3) + 2V / (V/π)^(1/3). Simplifying further, A = 2π(V/π)^(2/3) + 2π(V/π)^(1/3).
Combining like terms, A = 2π(V/π)^(1/3)(1 + (V/π)^(1/3)). Now, let's divide the surface area by the volume to find the ratio of the height to the diameter. We have (A/V) = (2π(V/π)^(1/3)(1 + (V/π)^(1/3))) / V, which simplifies to (A/V) = 2π(V/π)^(1/3) / V + (V/π)^(1/3) / V. Further simplifying, (A/V) = 2(1/π)^(1/3) + (1/π)^(1/3). To express this ratio in terms of π, we can multiply both the numerator and the denominator by (π/π)^(1/3), giving us (A/V) = 2π^(1/3) + π^(1/3). Rearranging, we have (A/V) = (π+2π^(1/3)) / π and the ratio of the height of the cylinder to the diameter of its semicircular ends is π / (π+2π^(1/3)).