smc

Because the arch has a height of $16$ inches, an equation that models the arch is $y=a{x}^2+16$, where $x$ is the horizontal distance from the center and $y$ is the height. The arch has a span of $40$ inches, so the arch meets the ground $20$ inches from the center. That means $0=400a+16$, so $a=-\frac{1}{25}$. Thus, the equation that models height based on distance from the center $y=-\frac{1}{25}{x}^2+16$, so the height of the arch $5$ inches from the center is $-\frac{1}{25}\cdot 5^2+16=\boxed{15}$ inches.