Find the area of a sector of a circle with radius 4 cm and of angle 30 degrees. Also, find the area of the corresponding major sector. Use pi = 3.14.

Step 1: Identify Given Values

We are given the radius of the circle $r = 4$ cm and the angle of the minor sector $\theta = 30^\circ$. We are also instructed to use $\pi = 3.14$ for calculations.

Step 2: Calculate Area of Minor Sector

The formula for the area of a sector with angle $\theta$ (in degrees) and radius $r$ is $\frac{\theta}{360^\circ} \times \pi r^2$. We will substitute the given values into this formula to find the area of the minor sector.

Step 3: Substitute Values and Calculate Minor Sector Area

Substitute the values of $\theta = 30^\circ$, $r = 4$ cm, and $\pi = 3.14$ into the formula. First, simplify the fraction $\frac{30}{360}$ to $\frac{1}{12}$. Then, calculate $4^2 = 16$. Finally, multiply all values to get the sector area.

Step 4: Perform Calculation for Minor Sector

Perform the multiplication: $\frac{1}{12} \times 3.14 \times 16 = \frac{50.24}{12}$. Dividing $50.24$ by $12$ yields approximately $4.1866...$ cm $^2$. Rounding to two decimal places, this is $4.19$ cm $^2$.

Step 5: Calculate Angle of Major Sector

The total angle in a circle is $360^\circ$. To find the angle of the major sector, we subtract the angle of the minor sector from $360^\circ$. This gives us $360^\circ - 30^\circ = 330^\circ$.

Step 6: Calculate Area of Major Sector

We can calculate the area of the major sector using the same sector area formula, but with the major sector angle of $330^\circ$. Substitute $\theta = 330^\circ$, $r = 4$ cm, and $\pi = 3.14$ into the formula.

Step 7: Substitute Values and Calculate Major Sector Area

Substitute the values: $\frac{330}{360} \times 3.14 \times 16$. Simplify the fraction $\frac{330}{360}$ to $\frac{11}{12}$. Then perform the multiplication: $\frac{11}{12} \times 3.14 \times 16 = \frac{11 \times 50.24}{12} = \frac{552.64}{12} \approx 46.05\text{ cm}^2$.

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