10. Given that $x$ is an acute angle, the minimum value of $\frac{8}{\sin x}+\frac{1}{\cos x}$ is $\qquad$
Applying the power mean inequality, we have
$$
\frac{8}{\sin x}+\frac{1}{\cos x}=\frac{4^{\frac{3}{2}}}{\left(\sin ^{2} x\right)^{\frac{1}{2}}}+\frac{1^{\frac{3}{2}}}{\left(\cos ^{2} x\right)^{\frac{1}{2}}} \geqslant \frac{(4+1)^{\frac{3}{2}}}{\left(\sin ^{2} x+\cos ^{2} x\right)^{\frac{1}{2}}}=5^{\frac{3}{2}}=5 \sqrt{5} \text {, }
$$

equality holds when $\frac{4}{\sin ^{2} x}=\frac{1}{\cos ^{2} x} \Rightarrow \sin x=\frac{2}{\sqrt{5}}, \cos x=\frac{1}{\sqrt{5}}$.
Therefore, the minimum value of $\frac{8}{\sin x}+\frac{1}{\cos x}$ is $5 \sqrt{5}$.