A circular decorative window has a radius of 18 cm. A straight frame element connects two boundary points such that it forms an equilateral triangle together with the radii drawn to those points from the central hub. If the smaller circular segment created by this frame is coated with a tinted film, approximately what percentage of the window's total area is tinted?

The correct option is 3%.

To find the percentage of the window's total area coated with tinted film, we need to calculate the area of the smaller circular segment created by the frame element and divide it by the total area of the circular window.

Step 1: Understand the Geometry
The circular window has a radius _{$r=18\text{cm}$}.
Two radii and a straight frame element form an equilateral triangle. Since all interior angles of an equilateral triangle are equal to 60°, the central angle subtended by the circular sector is _{$\theta =60°$}.

Step 2: Calculate the Total Area of the Window

${\text{Area}}_{\text{total}}=\pi r^2=\pi {\left(18\right)}^2=324\pi \approx 1017.88{\text{cm}}^2$

Step 3: Calculate the Area of the Circular Sector

${\text{Area}}_{\text{sector}}=\frac{60°}{360°}\times \pi r^2=\frac{1}{6}\times 324\pi =54\pi \approx 169.65{\text{cm}}^2$

Step 4: Calculate the Area of the Equilateral Triangle

${\text{Area}}_{\text{triangle}}=\frac{\sqrt{3}}{4}{r}^2=\frac{\sqrt{3}}{4}{\left(18\right)}^2=81\sqrt{3}\approx 140.30{\text{cm}}^2$

Step 5: Calculate the Tinted Segment Area

${\text{Area}}_{\text{segment}}={\text{Area}}_{\text{sector}}-{\text{Area}}_{\text{triangle}}=54\pi -81\sqrt{3}\approx 169.65-140.30=29.35{\text{cm}}^2$

Step 6: Determine the Percentage of Total Area Tinted

$\text{Percentage}=\left(\frac{{\text{Area}}_{\text{segment}}}{{\text{Area}}_{\text{total}}}\right)\times 100\%$

$\text{Percentage}\approx \left(\frac{29.35}{1017.88}\right)\times 100\%\approx 2.88\%$

Rounding to the nearest whole number, approximately 3% of the window's total area is tinted.