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Difficulty: HardCircles

Match each advanced circular geometry scenario with its exact calculated numerical value.

  • A wheel with a diameter of 2424 inches rolls forward for a total distance of 120π120\pi inches. The number of complete revolutions the wheel makes.55
  • The measure of the central angle (in degrees) of a circular sector that has a radius of 66 inches and an area of 15π15\pi square inches.150150
  • The outer circumference (in yards) of a circular track that has an inner circumference of 40π40\pi yards and a uniform width of 55 yards.50π50\pi
  • The area (in square inches) of a square perfectly inscribed inside a circular base that has an area of 64π64\pi square inches.128128

Answer

Match the scenarios to their respective values: the wheel makes 55 revolutions, the sector angle is 150∘150^{\circ}, the track's outer circumference is 50π50\pi yards, and the inscribed square's area is 128128 square inches.
Each scenario relies on accurately applying distinct circular properties. 1) The distance traveled by a wheel uses the circumference formula (C=πdC=\pi d). 2) The sector area requires using the proportion of the circle θ360πr2\frac{\theta}{360}\pi r^2. 3) Concentric track perimeters require adding the width to the inner radius before recalculating circumference. 4) An inscribed square has a diagonal equal to the circle's diameter, allowing for area calculation via the diagonal.

Step-by-Step Solution

1
Calculate the number of wheel revolutions.
The circumference is 24π24\pi. Dividing the total distance 120π120\pi by 24π24\pi yields 55 revolutions.
One full revolution covers a linear distance equal to exactly one circumference.
2
Determine the sector's central angle.
Using the area formula A=θ360πr2A = \frac{\theta}{360}\pi r^2, setting 15π=θ360π(36)15\pi = \frac{\theta}{360}\pi(36) simplifies to θ=150∘\theta = 150^{\circ}.
The area of a sector is a proportional fraction of the full circle's area based on its central angle.
3
Find the outer circumference of the circular track.
The inner radius is 2020 (from 40π=2πr40\pi = 2\pi r). The outer radius is 20+5=2520 + 5 = 25. The outer circumference is 2π(25)=50π2\pi(25) = 50\pi.
Concentric circles share the same center; the outer radius equals the inner radius plus the track's width.
4
Calculate the area of the inscribed square.
The circle's radius is 88, so the diameter (and square's diagonal) is 1616. The square's area is 12(16)2=128\frac{1}{2}(16)^2 = 128.
An inscribed square's diagonal is strictly equal to the diameter of the circumscribing circle.

Key Concept

Advanced applications of circle formulas including circumference, sector area, and relationships with inscribed shapes.
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Topics

Circles

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