The volume \( V \) of a cone can be calculated using the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone.
In this case, the radius \( r \) is 5 inches and the height \( h \) is 15 inches. Plugging the values into the formula:
\[ V = \frac{1}{3} \times 3.14 \times (5)^2 \times 15 \]
First, calculate \( (5)^2 \):
\[ (5)^2 = 25 \]
Now plug this value into the formula:
\[ V = \frac{1}{3} \times 3.14 \times 25 \times 15 \]
Calculate \( 25 \times 15 \):
\[ 25 \times 15 = 375 \]
Now substitute this back into the volume formula:
\[ V = \frac{1}{3} \times 3.14 \times 375 \]
Calculate \( 3.14 \times 375 \):
\[ 3.14 \times 375 = 1177.5 \]
Now, divide by 3:
\[ V = \frac{1177.5}{3} = 392.5 \]
Thus, the volume of water that the flower vase can hold is:
\[ \boxed{392.5 \text{ cubic inches}} \]