In this activity, you are going to explore the Texas Star ferris wheel in Dallas, TX. We have provided the diameter and number of capsules below to get you started. Complete the following tasks and calculations.
Diameter: 212 feet
Number of compartments: 44
1. Circumference of the wheel (in feet)
2. Area of the wheel (in feet)
3. Measure of a Central angle in degrees
4. Arc length between two cars or compartments
5. Area of a sector between two cars or compartments
6. If a smaller replica of the Texas Star ferris wheel was constructed, what conclusions could you draw about the central angle of the original wheel and replica?
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To solve this math problem, we can use the formulas for circumference, area, central angle, arc length, and sector area.
1. Circumference of the wheel:
The formula to calculate the circumference of a circle is C = πd, where d is the diameter.
C = π * 212 ft
C = 665.68 ft
2. Area of the wheel:
The formula to calculate the area of a circle is A = πr^2, where r is the radius (half the diameter).
r = 212 ft / 2
r = 106 ft
A = π * (106 ft)^2
A = 35412.14 ft^2
3. Measure of a Central angle in degrees:
The total central angle in a circle is always 360 degrees.
Each compartment is a portion of the total central angle, evenly spaced around the wheel.
Central angle in degrees = 360 degrees / number of compartments
Central angle in degrees = 360 degrees / 44
Central angle in degrees ≈ 8.18 degrees
4. Arc length between two cars or compartments:
The formula to calculate the arc length is L = (Central angle in radians) * radius, where L is the arc length in radians.
First, we convert the central angle from degrees to radians.
Central angle in radians = Central angle in degrees * (π / 180 degrees)
Central angle in radians ≈ 8.18 degrees * (π / 180 degrees)
Central angle in radians ≈ 0.142 radians
L = 0.142 radians * 106 ft
L ≈ 15.01 ft
5. Area of a sector between two cars or compartments:
The formula to calculate the area of a sector is A = (Central angle in radians / 2) * (radius^2), where A is the area of the sector in square feet.
A = (0.142 radians / 2) * (106 ft)^2
A ≈ 756.24 ft^2
6. If a smaller replica of the Texas Star ferris wheel was constructed, the central angle of the original wheel and replica would be proportional. This means that if the replica has a smaller diameter, the central angle would also be smaller, and vice versa. The relationship would be linear, as long as the proportions and dimensions of the replica are maintained.
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