Step by step on how to find the length of arc ACB and the area of sector OACB for each given value of theta:
1. Knowing the Circumference:
We are given that the circle's circumference is 88 cm. We can use the formula for circumference to find the radius:
`circumference = 2 * pi * radius`
Therefore, `radius = circumference / (2 * pi) = 88 cm / (2 * pi) ≈ 14 cm`.
2. Calculating Arc Length and Sector Area:
For each value of theta, we can follow these steps:
- Convert theta to radians: Multiply theta by pi and divide by 180 degrees.
- Ratio:** Calculate the ratio of theta to 360 degrees (theta / 360).
- Arc Length (ACB): Multiply the ratio by the circle's circumference to find the arc length.
- Sector Area (OACB): Multiply the ratio by half of pi and the square of the radius, then divide by 2.
3. Solving for each value of theta:
a) theta = 60 degrees:
- Theta in radians = (60 * pi) / 180 ≈ 1.047 radians
- Ratio = 60 / 360 = 1/6
- Arc Length (ACB) = (1/6) * 88 cm ≈ 14.67 cm
- Sector Area (OACB) = (1/6) * (pi/2) * (14 cm)^2 ≈ 506.84 cm^2
**b) theta = 90 degrees:
- Theta in radians = (90 * pi) / 180 ≈ 1.571 radians
- Ratio = 90 / 360 = 1/4
- Arc Length (ACB) = (1/4) * 88 cm ≈ 22.00 cm
- Sector Area (OACB) = (1/4) * (pi/2) * (14 cm)^2 ≈ 760.27 cm^2
c) theta = 120 degrees:
- Theta in radians = (120 * pi) / 180 ≈ 2.094 radians
- Ratio = 120 / 360 = 1/3
- Arc Length (ACB) = (1/3) * 88 cm ≈ 29.33 cm
- Sector Area (OACB) = (1/3) * (pi/2) * (14 cm)^2 ≈ 1013.69 cm^2
d) theta = 210 degrees:
- Theta in radians = (210 * pi) / 180 ≈ 3.665 radians
- Ratio = 210 / 360 = 7/12
- Arc Length (ACB) = (7/12) * 88 cm ≈ 51.33 cm
- Sector Area (OACB) = (7/12) * (pi/2) * (14 cm)^2 ≈ 1773.95 cm^2
Therefore, for each given value of theta, you can calculate the arc length and sector area using the abovementioned formulas and steps.
I try to recalculate the result and see how python mathpotlib create graph for it
