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Difficulty: Very hardCircles

Match each circular geometry scenario with its corresponding simplified algebraic expression.

  • The area of a circular pathway of uniform width xx surrounding a pool with radius rrπx(2r+x)\pi x(2r + x)
  • The total perimeter of a semicircular window with radius rrr(π+2)r(\pi + 2)
  • The number of full revolutions a tire with diameter dd makes to travel a distance DDDπd\frac{D}{\pi d}
  • The radius of a circle whose area is exactly half the area of a circle with radius RRR22\frac{R\sqrt{2}}{2}

Answer

The correct pairings connect each geometric scenario to its simplified algebraic derivation by applying circle formulas, expanding binomials, and rationalizing denominators.
Each expression is correctly derived by applying the fundamental formulas for circle area and circumference, followed by rigorous algebraic simplification such as factoring, binomial expansion, or radical rationalization.

Step-by-Step Solution

1
Analyze the annular pathway area scenario.
πx(2r+x)\pi x(2r + x)
Requires subtracting the inner pool area from the total outer area and expanding the squared binomial (r+x)2(r+x)^2.
2
Determine the semicircular perimeter expression.
r(π+2)r(\pi + 2)
Requires adding the semicircular arc length to the straight diameter base and factoring out the radius.
3
Calculate the algebraic expression for tire revolutions.
Dπd\frac{D}{\pi d}
Requires dividing the total linear distance by the circumference of the tire, utilizing the diameter.
4
Solve for the half-area radius.
R22\frac{R\sqrt{2}}{2}
Requires setting up an area ratio equation, solving for the new radius, and rationalizing the denominator of the resulting square root.

Key Concept

Algebraic representation and derivation of advanced circular measurements
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Topics

Circles

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