A rectangle's length and width are increased by 10%. By what percentage does its area increase?

Let the original length of the rectangle be $L$ and the original width be $W$.
The original area, $A_1$, is given by:

The sides are increased by 10%.
New length, $L'$, is $L + 0.10L = 1.10L$.
New width, $W'$, is $W + 0.10W = 1.10W$.

The new area, $A_2$, is given by:

The percentage change is given by the formula:

Substitute the expressions for $A_1$ and $A_2$:

The percentage change in its area is \boxed{\text{21%}}.

15.

Let the two-digit number be $10t + u$, where $t$ is the tens digit and $u$ is the units digit.

"the sum of its digits is 9"

"When the digits are reversed, the new number exceeds the old number by 45."
The new number with digits reversed is $10u + t$.
The old number is $10t + u$.
So, the difference is:

Divide by 9:

We have:

1. $t + u = 9$
2. $-t + u = 5$

Add Equation 1 and Equation 2:

Substitute $u = 7$ into Equation 1:

The original number is $10t + u$.

The original number is $\boxed{27}$.

16.

The volume of a prism is given by the area of its cross-section multiplied by its length.

A regular hexagon can be divided into 6 equilateral triangles. The side length of the hexagon is $s = 5$ cm.
The area of one equilateral triangle with side $s$ is $\frac{\sqrt{3}}{4}s^2$.
The area of the regular hexagonal base, $A_{base}$, is:

Substitute $s = 5$ cm:

The length of the prism, $L$, is 12 cm.

The volume of the prism is $\boxed{450\sqrt{3} \text{ cm}^3}$.