a. A sector that has been cut out of circular plastic sheet which has a circumference of 12.31 m. If the central angle of the sector is 1/4 of a revolution, find the area of the sector.
This is not trig.
Anyway, C=2(pi)r. If the circumference is 12.31, then the radius is 12.31/[(pi)r] = 1.96. If the central angle is 1/4 of a revolution, then the central angle measure is 90 degrees.
The area of a sector is a piece of the area of the entire circle:
A=(pi)r^2 * central angle measure/360
A=(pi)1.96^2 * 90/360
A=3 m^2
To find the area of the sector, we need two pieces of information: the central angle and the circumference of the circular plastic sheet.
Let's start by finding the central angle. We are given that the central angle is 1/4 of a revolution. In a full revolution, there are 360 degrees. Therefore, 1/4 of a revolution is equal to (1/4) * 360 = 90 degrees.
Next, we need to find the radius of the circular plastic sheet. The circumference of a circle is given by the formula:
circumference = 2πr,
where r is the radius of the circle. Rearranging the formula to solve for r, we get:
r = circumference / (2π).
Substituting the given value of the circumference (12.31 m), we have:
r = 12.31 / (2 * 3.14159) ≈ 1.9589 m.
Now that we have the radius, we can calculate the area of the sector. The formula for the area of a sector is:
Area = (central angle / 360) * π * r^2.
Plugging in the values, we get:
Area = (90 / 360) * 3.14159 * (1.9589^2) ≈ 1.4431 square meters.
Therefore, the area of the sector is approximately 1.4431 square meters.