Set up variables

Since the ratio of radius $r$ to height $h$ is $3: 4$, let $r = 3k$ and $h = 4k$.

$r = 3k, \quad h = 4k$

Apply volume formula

Substitute the variables into the cylinder volume formula: $V = \pi r^2 h = 38808$.

$\frac{22}{7} \times (3k)^2 \times (4k) = 38808$

Solve for k

Simplify: $\frac{22}{7} \times 9k^2 \times 4k = 38808 \Rightarrow \frac{792}{7} k^3 = 38808$. Then $k^3 = 38808 \times \frac{7}{792} = 49 \times 7 = 343$.

$k = \sqrt[3]{343} = 7$

Calculate diameter

Calculate radius $r = 3k = 3(7) = 21$ cm. The diameter $d = 2r = 2(21) = 42$ cm.

$d = 2 \times 21 = 42 \text{ cm}$