Question

what is the exact area of the shaded sector in the circle shown below? 50° c 12m 20π m² 12π m² 5π m² (5/3)π m²

Explanation:

Step1: Find the radius of the circle

The diameter of the circle is given as \( 12 \, \text{m} \), so the radius \( r \) is half of the diameter.
\( r=\frac{12}{2} = 6 \, \text{m} \)

Step2: Recall the formula for the area of a sector

The formula for the area of a sector with central angle \( \theta \) (in degrees) and radius \( r \) is \( A=\frac{\theta}{360^{\circ}}\times\pi r^{2} \)

Step3: Substitute the values into the formula

Here, \( \theta = 50^{\circ} \) and \( r = 6 \, \text{m} \)
\( A=\frac{50^{\circ}}{360^{\circ}}\times\pi\times(6)^{2} \)
First, calculate \( (6)^{2}=36 \)
Then, \( \frac{50}{360}\times\pi\times36 \)
Simplify \( \frac{50}{360}\times36=\frac{50\times36}{360}=5 \)
So, \( A = 5\pi \, \text{m}^2 \)

Wait, there seems to be a mistake above. Let's recalculate:

Wait, the diameter is 12m, so radius \( r = 6 \)m. The formula for the area of a sector is \( A=\frac{\theta}{360}\times\pi r^{2} \)

Substituting \( \theta = 50^{\circ} \), \( r = 6 \):

\( A=\frac{50}{360}\times\pi\times6^{2}=\frac{50}{360}\times\pi\times36=\frac{50\times36\pi}{360}=\frac{50\pi}{10} = 5\pi \)? Wait, no, 36 divided by 360 is 0.1, 50 times 0.1 is 5? Wait, no, 36/360 = 1/10, so 50*(1/10)=5, so 5π? But let's check the options. Wait, maybe I misread the diameter. Wait, the line from C to the bottom is 12m, so diameter is 12m, radius 6m. Wait, but let's check the options. The options include 20π, 12π, 5π, 5/3 π. Wait, maybe I made a mistake in the radius. Wait, maybe the 12m is the radius? Wait, the diagram shows a line from C to a point on the circle as 12m? Wait, maybe the diameter is 12m, so radius 6m. Wait, let's re-express the formula:

Wait, if the radius is 6m, then \( r^2 = 36 \), \( \frac{50}{360} \times \pi \times 36 = \frac{50 \times 36 \pi}{360} = 5\pi \). But let's check the options. One of the options is 20π? Wait, maybe the radius is 12m? Wait, maybe the 12m is the radius. Let's try that. If radius \( r = 12 \)m, then:

\( A=\frac{50}{360}\times\pi\times(12)^2=\frac{50}{360}\times\pi\times144=\frac{50\times144\pi}{360}=\frac{7200\pi}{360}=20\pi \)

Ah! That must be it. I misread the diagram. The line from C to the point on the circle is the radius, so radius \( r = 12 \)m, not the diameter. Let's correct that.

Step1: Determine the radius

From the diagram, the length from the center C to a point on the circle is 12m, so the radius \( r = 12 \, \text{m} \)

Step2: Use the sector area formula

The formula for the area of a sector is \( A=\frac{\theta}{360^{\circ}}\times\pi r^{2} \), where \( \theta = 50^{\circ} \) and \( r = 12 \, \text{m} \)

Step3: Substitute the values

\( A=\frac{50^{\circ}}{360^{\circ}}\times\pi\times(12)^{2} \)
First, calculate \( (12)^{2}=144 \)
Then, \( \frac{50}{360}\times\pi\times144 \)
Simplify \( \frac{50\times144}{360}=\frac{7200}{360}=20 \)
So, \( A = 20\pi \, \text{m}^2 \)

Answer:

\( 20\pi \, \text{m}^2 \) (Corresponding to the option: 20π m²)