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.
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Step-by-Step Solution
Step 1: Identify Given Values
We are given the radius of the circle r=4 cm and the angle of the minor sector θ=30∘. We are also instructed to use π=3.14 for calculations.
Step 2: Calculate Area of Minor Sector
The formula for the area of a sector with angle θ (in degrees) and radius r is 360∘θ×πr2. 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 θ=30∘, r=4 cm, and π=3.14 into the formula. First, simplify the fraction 36030 to 121. Then, calculate 42=16. Finally, multiply all values to get the sector area.
Step 4: Perform Calculation for Minor Sector
Perform the multiplication: 121×3.14×16=1250.24. 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∘. To find the angle of the major sector, we subtract the angle of the minor sector from 360∘. This gives us 360∘−30∘=330∘.
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∘. Substitute θ=330∘, r=4 cm, and π=3.14 into the formula.
Step 7: Substitute Values and Calculate Major Sector Area
Substitute the values: 360330×3.14×16. Simplify the fraction 360330 to 1211. Then perform the multiplication: 1211×3.14×16=1211×50.24=12552.64≈46.05 cm2.