Let's start by finding the ratios of the volumes of shapes B and C. If the ratio of their volumes is 27:64, then we can write:
Volume of B/Volume of C = 27/64
Let's call the height of shape B "hB" and the height of shape C "hC". Since the shapes are similar, their dimensions are proportional.
We can set up the following ratio:
hB/hC = √(25/4) (square root of the ratio of the surface areas of B and A)
Simplifying this gives:
hB/hC = 5/2
Now we need to express the ratio of the height of shape A to the height of shape C in terms of hB and hC:
height of A/height of C = √(surface area of A/surface area of C) = √(4/surface area of C)
Next, we need to express the surface area of shape C in terms of its height hC:
Surface area of C = 2πr² + 2πrhC
We know that shapes A, B, and C are similar, so their dimensions are proportional. Let's call the radius of shape A "rA". Then we can write:
rA/rB = hA/hB
rA/rC = hA/hC
Since the shapes are solid, we know that their volumes are proportional to the cubes of their dimensions:
Volume of A/Volume of B = (hA/hB)³
Volume of A/Volume of C = (hA/hC)³
Now we can substitute for rA and simplify:
Surface area of C = 2π(hA/hC)²rC + 2πrhC
Surface area of C = 2πhC²(rA/rC) + 2πrhC
Surface area of C = 2πhC²(hA/hC)² + 2πrhC
Surface area of C = 2πhC²(hA²/hC²) + 2πrhC
Surface area of C = 2πhA²(rC/hB)² + 2π(rA/hB)(hA/hC)hC
Surface area of C = 2πhA²(hC/hB)² + 2π(hA/hB)(rB/rC)hC
Surface area of C = 2πhA²(hC/hB)² + 2π(hA/hB)(hB/hC)hC
Surface area of C = 2πhA²(hC/hB)² + 2πhA(hC/hC)
Surface area of C = 2πhA²(hC/hB)² + 2πhA
Now we can substitute in the values we know:
4/surface area of C = 4/(2πhA²(hC/hB)² + 2πhA) = (2/πhA)(hC/hB)² + 1/2
Finally, we can express the ratio of the height of shape A to the height of shape C in terms of hB and hC:
height of A/height of C = √(surface area of A/surface area of C) = √(4/surface area of C)
height of A/height of C = √(4/[(2/πhA)(hC/hB)² + 1/2])
Simplifying this gives:
height of A/height of C = √(8πhB²/[4πhA²(hC/hB)² + πhA])
height of A/height of C = √(2hB²/[hA²(hC/hB)² + 1/2hA])
height of A/height of C = √(2*5²/[(2/5)²*(2/27) + 1/8])
height of A/height of C = √(2*25/[(4/135) + 1/8])
height of A/height of C = √(5400/61)
So the ratio of the height of shape A to the height of shape C is √(5400/61):1, which is the simplest form.