Answer
Area of red zone is $\frac{17}{4}\pi {{\text{m}}^{2}}$
Work Step by Step
Compare the outer and inner circle equation with the standard equation of a circle.
${{\left( x-h \right)}^{2}}+{{\left( y-k \right)}^{2}}={{r}^{2}}$
For outer circle radius it would be ${{r}_{o}}^{2}=\frac{81}{4}$.
And for the inner circle it would be ${{r}_{i}}^{2}=16$
Use the formula for Area of a circle to calculate the red zone area, $A=\pi {{r}^{2}}$.
The area of red zone is the difference between the area of outer and inner circles.
Put the value of outer and inner radius in the formula, $A=\pi {{r}^{2}}$
$\begin{align}
& \text{Area of red zone}=\pi {{r}_{o}}^{2}-\pi {{r}_{i}}^{2} \\
& =\left( \pi \times \frac{81}{4} \right)-\left( \pi \times 16 \right) \\
& =\pi \left( \frac{81-64}{4} \right) \\
& =\frac{17}{4}\pi {{\text{m}}^{2}}
\end{align}$
Thus, the area of red zone is $\frac{17}{4}\pi {{\text{m}}^{2}}$.
