This question tests understanding of the relationship between arc length, radius, and central angle in radians. The arc length s of a sector with central angle θ (in radians) and radius r is given by s = rθ, which derives from the fact that arc length is proportional to the central angle: the ratio θ/(2π) equals the ratio s/(2πr). Using the formula s = rθ with r = 6 cm, the shorter arc (θ = π/4) has length s₁ = 6 × (π/4) = 3π/2 cm, and the longer arc (θ = π/2) has length s₂ = 6 × (π/2) = 3π cm. Choice B is correct because the ratio (longer arc):(shorter arc) = 3π : (3π/2) = 2:1, showing that doubling the angle doubles the arc length. Choice A incorrectly reverses the ratio, giving (shorter arc):(longer arc) instead. When working with radians, arc lengths are directly proportional to their central angles when the radius is constant.

This question tests understanding of the relationship between arc length, radius, and central angle in radians. The arc length s of a sector with central angle θ (in radians) and radius r is given by s = rθ, which derives from the fact that arc length is proportional to the central angle: the ratio θ/(2π) equals the ratio s/(2πr). Using the formula s = rθ, where r = 10 cm and θ = π/6 radians, we calculate s = 10 × (π/6) = 10π/6 cm. Choice A is correct because it properly applies the arc length formula to get 10π/6 cm. Choice C incorrectly simplifies 10π/6 to 10π/3, which would double the actual arc length by reducing the denominator from 6 to 3. Key to radian problems: always identify the radius first, then use s = rθ remembering that θ must be in radians, not degrees.

This question tests understanding of the relationship between arc length, radius, and central angle in radians. The arc length s of a sector with central angle θ (in radians) and radius r is given by s = rθ, which derives from the fact that arc length is proportional to the central angle: the ratio θ/(2π) equals the ratio s/(2πr). The key insight is that when r = 1, the formula s = rθ simplifies to s = θ, meaning the numerical value of the angle in radians equals the numerical value of the arc length. Choice D is correct because it connects to the stimulus data and shows correct application of s = rθ with specific values, where doubling r multiplies s by 2. Choice C incorrectly assumes the area proportionality, multiplying by 4 instead of recognizing the linear relationship for arc length. Remember that on a unit circle, radian measure and arc length are numerically equal because r = 1; on other circles, you must account for the radius. The formula s = rθ only works when θ is in radians; if given degrees, you must convert first using θ(radians) = (π/180)θ(degrees).

This question tests understanding of the relationship between arc length, radius, and central angle in radians. The radian measure is defined as the ratio of arc length to radius: θ = s/r, which means that for a given angle, arc length is directly proportional to radius with the angle (in radians) as the constant of proportionality. Using the definition θ = s/r, where s = 10 and r = 5, we find θ = 10/5 = 2 radians. Choice B is correct because it connects to the stimulus data and shows correct application of θ = s/r with specific values. Choice C incorrectly treats the radian measure as degrees, when radians are a different unit of angular measurement based on arc length. Key to radian problems: always identify the radius first, then use s = rθ remembering that θ must be in radians, not degrees. To check your understanding: one complete revolution around any circle is 2π radians because the circumference (2πr) divided by the radius (r) equals 2π.

This question tests understanding of the relationship between arc length, radius, and central angle in radians, and the derivation and application of the sector area formula. The area of a sector with central angle θ (in radians) and radius r is A = (1/2)r²θ, derived from the fact that a sector occupies the fraction θ/(2π) of the total circle area πr². Using the formula A = (1/2)r²θ, where r = 4 and θ = π/2 radians, we calculate A = (1/2)16(π/2) = 8*(π/2) = 4π cm². Choice B is correct because it connects to the stimulus data and shows correct application of A = (1/2)r²θ with specific values. Choice D incorrectly forgets the 1/2 and the θ/2, using something like πr² fraction without proper scaling. Key to radian problems: always identify the radius first, then use s = rθ remembering that θ must be in radians, not degrees. Both the arc length formula s = rθ and the sector area formula A = (1/2)r²θ require the angle to be measured in radians because radian measure is defined as the dimensionless ratio s/r.

This question tests understanding of the relationship between arc length, radius, and central angle in radians. The arc length s of a sector with central angle θ (in radians) and radius r is given by s = rθ, which derives from the fact that arc length is proportional to the central angle: the ratio θ/(2π) equals the ratio s/(2πr). Using the formula s = rθ, where r = 12 and θ = 2π/3 radians, we calculate s = 12 * (2π/3) = 24π/3 = 8π meters. Choice A is correct because it connects to the stimulus data and shows correct application of s = rθ with specific values. Choice C incorrectly uses the circumference formula 2πr instead of the arc length formula rθ. Key to radian problems: always identify the radius first, then use s = rθ remembering that θ must be in radians, not degrees. The formula s = rθ only works when θ is in radians; if given degrees, you must convert first using θ(radians) = (π/180)θ(degrees).

This question tests understanding of the relationship between arc length, radius, and central angle in radians. The arc length s of a sector with central angle θ (in radians) and radius r is given by s = rθ, which derives from the fact that arc length is proportional to the central angle: the ratio θ/(2π) equals the ratio s/(2πr). Since both angles are measured on circles, the ratio of arc length to radius must equal the angle in radians for each: for r=5, s=5*(π/3); for r=10, s=10*(π/3), which is twice the original. Choice D is correct because it connects to the stimulus data and shows correct application of s = rθ with specific values. Choice B incorrectly assumes arc length scales with the square of the radius, confusing it with area. Remember that on a unit circle, radian measure and arc length are numerically equal because r = 1; on other circles, you must account for the radius. The formula s = rθ only works when θ is in radians; if given degrees, you must convert first using θ(radians) = (π/180)θ(degrees).

This question tests understanding of the relationship between arc length, radius, and central angle in radians, and the derivation and application of the sector area formula. Both the arc length formula s = rθ and the sector area formula A = (1/2)r²θ require the angle to be measured in radians because radian measure is defined as the dimensionless ratio s/r. Using the formulas, where r = 3 and θ = 3π/4 radians, we calculate s = 3*(3π/4) = 9π/4 m and A = (1/2)9(3π/4) = (9/2)*(3π/4) = 27π/8 m². Choice A is correct because it connects to the stimulus data and shows correct application of s = rθ and A = (1/2)r²θ with specific values. Choice C incorrectly doubles the area by forgetting the 1/2 in the sector formula. Key to radian problems: always identify the radius first, then use s = rθ remembering that θ must be in radians, not degrees. When working with radians, express answers in terms of π rather than decimal approximations unless the context specifically requires decimals.

This question tests understanding of the relationship between arc length, radius, and central angle in radians. The radian measure is defined as the ratio of arc length to radius: θ = s/r, which means that for a given angle, arc length is directly proportional to radius with the angle (in radians) as the constant of proportionality. Using the definition θ = s/r, where s = 4π and r = 8, we find θ = 4π/8 = π/2 radians, which is the ratio asked. Choice B is correct because it connects to the stimulus data and shows correct application of θ = s/r with specific values. Choice C incorrectly uses the circumference formula 2πr instead of the arc length formula rθ. Remember that on a unit circle, radian measure and arc length are numerically equal because r = 1; on other circles, you must account for the radius. To check your understanding: one complete revolution around any circle is 2π radians because the circumference (2πr) divided by the radius (r) equals 2π.

This question tests understanding of the relationship between arc length, radius, and central angle in radians. The arc length s of a sector with central angle θ (in radians) and radius r is given by s = rθ, which derives from the fact that arc length is proportional to the central angle: the ratio θ/(2π) equals the ratio s/(2πr). Using the formula s = rθ, where r = 3 and θ = π radians, we calculate s = 3π m. Choice A is correct because it connects to the stimulus data and shows correct application of s = rθ with specific values. Choice B incorrectly uses diameter instead of radius in the calculation, doubling the correct answer. Key to radian problems: always identify the radius first, then use s = rθ remembering that θ must be in radians, not degrees. To check your understanding: one complete revolution around any circle is 2π radians because the circumference (2πr) divided by the radius (r) equals 2π.