Question

1. which of the following is closest to the area shaded in green in the image?

image of a circular diagram with a blue inner circle (radius 4), a red ring around it, and a green shaded sector with a central angle of 70 degrees. the outer radius (from center to outer edge of red) is 4 + 6 = 10? wait, the diagram has a blue circle with radius 4, then a red ring, and the green sector is between radius 4 and radius 4 + 6 = 10? the central angle is 70 degrees. the options are 1. 48, 2. 49, 3. 50, 4. 51, 5. 52, 6. 53, 7. 54, 8. 55, 9. 56, 10. 57 (partially visible)

Explanation:

Step1: Identify the shape and formula

The shaded area is a sector of an annulus (ring - shaped region). The formula for the area of a sector of an annulus is $A=\frac{\theta}{360}\times\pi\times(R^{2}-r^{2})$, where $\theta$ is the central angle in degrees, $R$ is the outer radius, and $r$ is the inner radius. Here, $\theta = 70^{\circ}$, $r = 4$, and $R=4 + 6=10$.

Step2: Substitute the values into the formula

First, calculate $R^{2}-r^{2}$: $R^{2}-r^{2}=10^{2}-4^{2}=100 - 16 = 84$.
Then, substitute into the sector area formula: $A=\frac{70}{360}\times\pi\times84$.

Step3: Simplify the expression

Simplify $\frac{70}{360}\times84=\frac{7}{36}\times84=\frac{7\times84}{36}=\frac{588}{36}=\frac{49}{3}\approx16.333$.
Then, $A=\pi\times\frac{49}{3}\approx3.1416\times\frac{49}{3}\approx\frac{153.9384}{3}\approx51.31$.

Answer:

1. 51