Zach, Adam, and Carlos are working on an art project. They each cut a sector out of different size circles. Details about the sectors and circles are given below. Sort the three sectors in order of increasing area. Place the sector with the smallest area on top.
Zach's sector has a radius of 5 cm and a central angle of π/4 radians. Adam's sector was cut from a circle with an area of 72 cm2 with a central angle of π/6 radians. Carlos' sector was cut from a circle with an area of 36π cm2, a circumference of 12π cm, and an arc length of 5 cm.
To sort the three sectors in order of increasing area, we need to calculate the area of each sector. Here's how to do it:
1. Zach's sector:
- Radius = 5 cm
- Central angle = π/4 radians
- Formula for the area of a sector = (π * r^2 * θ) / (2π)
- Substituting the values: Area = (π * 5^2 * π/4) / (2π) = (25π/4) / 2 = 25π/8 square cm
2. Adam's sector:
- Circle area = 72 cm^2
- Central angle = π/6 radians
- We need to find the radius of Adam's circle first.
- Formula for the area of a circle = π * r^2, where r is the radius
- Substituting the values: 72 = π * r^2
- Solving for r: r^2 = 72/π, so r = √(72/π)
- Now we can calculate the area of Adam's sector using the same formula as before.
- Area = (π * (√(72/π))^2 * π/6) / (2π) = (72/6) / 2 = 12 square cm
3. Carlos' sector:
- Circle area = 36π cm^2
- Circumference = 12π cm
- Arc length = 5 cm
- We need to find the radius of Carlos' circle first.
- Formula for the circumference of a circle = 2πr, where r is the radius
- Substituting the values: 12π = 2πr, so r = 12/2 = 6 cm
- Now we can calculate the area of Carlos' sector using the same formula as before.
- Area = (π * 6^2 * (5/12)) / (2π) = (180/12) / 2 = 15/2 square cm
So, the sectors in order of increasing area are as follows:
1. Adam's sector with an area of 12 square cm
2. Carlos' sector with an area of 15/2 (7.5) square cm
3. Zach's sector with an area of 25π/8 square cm