In the xy -plane, a line with equation 2 y = c for some constant c

The correct answer is $\frac{81}{4}$. The given linear equation is $2y=c$. Dividing both sides of this equation by $2$ yields $y=\frac{c}{2}$. Substituting $\frac{c}{2}$ for $y$ in the equation of the parabola yields $\frac{c}{2}=-2x^2+9x$. Adding $2x^2$ and $-9x$ to both sides of this equation yields $2x^2-9x+\frac{c}{2}=0$. Since it’s given that the line and the parabola intersect at exactly one point, the equation $2x^2-9x+\frac{c}{2}=0$ must have exactly one solution. An equation of the form $A{x}^2+Bx+C=0$, where $A$, $B$, and $C$ are constants, has exactly one solution when the discriminant, $B^2-4AC$, is equal to $0$. In the equation $2x^2-9x+\frac{c}{2}=0$, where $A=2$, $B=-9$, and $C=\frac{c}{2}$, the discriminant is ${\left(-9\right)}^2-4\left(2\right)\left(\frac{c}{2}\right)$. Setting the discriminant equal to $0$ yields ${\left(-9\right)}^2-4\left(2\right)\left(\frac{c}{2}\right)=0$, or $81-4c=0$. Adding $4c$ to both sides of this equation yields $81=4c$. Dividing both sides of this equation by $4$ yields $c=\frac{81}{4}$. Note that 81/4 and 20.25 are examples of ways to enter a correct answer.