In a circular design, the first curved section subtends a central angle measuring 144° with a radius of 7 cm. A second curved section, constructed with an identical radius, has a central angle measuring $\frac{4\pi }{5}$ radians. Determine the simplified ratio comparing the area of the first section to the area of the second section.

Correct Answer: 1:1

To find the ratio of the area of the first curved section (sector) to the area of the second section, we will calculate or compare the areas of both sectors step-by-step.

Step 1: Convert the central angle of the first section to radians (or both to degrees).
The first section has a central angle measuring 144°. We can convert this angle into radians using the conversion factor $\frac{\pi }{180}$ radians per degree:

${\theta }_1=144\times \frac{\pi }{180}=\frac{144\pi }{180}$

Simplifying the fraction by dividing the numerator and denominator by 36:

${\theta }_1=\frac{4\pi }{5}$ radians.

Step 2: Compare the central angles and radii of both sections.
The central angle of the first section is ${\theta }_1=\frac{4\pi }{5}$ radians.
The central angle of the second section is given as ${\theta }_2=\frac{4\pi }{5}$ radians.
Both sections also have an identical radius, $r=7$ cm.

Step 3: Calculate the area of a sector.
The area of a circular sector with radius $r$ and central angle $\theta$ in radians is given by the formula:

$\text{Area}=\frac{1}{2}{r}^2\theta$

Since both sectors have the exact same radius ($r=7$ cm) and the exact same central angle ($\theta =\frac{4\pi }{5}$ radians), their areas are equal:

${\text{Area}}_1={\text{Area}}_2$

Step 4: Find the ratio of the areas.
The ratio of the area of the first section to the area of the second section is:

$\frac{{\text{Area}}_1}{{\text{Area}}_2}=\frac{1}{1}$

Therefore, the simplified ratio comparing the area of the first section to the area of the second section is 1:1.