In Exercises 71–74, find the length of the arc on a circle of radius r intercepted by a central angle θ. Express arc length in terms of 𝜋. Then round your answer to two decimal places. Radius, r: 8 feet Central Angle, θ: θ = 225°

검증된 단계별 안내

Recall the formula for the length of an arc intercepted by a central angle \( \theta \) on a circle of radius \( r \): \[ \text{Arc length} = r \times \theta_{\text{radians}} \]

Convert the central angle from degrees to radians using the conversion factor \( \pi \text{ radians} = 180^\circ \): \[ \theta_{\text{radians}} = 225^\circ \times \frac{\pi}{180^\circ} \]

Simplify the radian measure by reducing the fraction: \[ \theta_{\text{radians}} = \frac{225}{180} \pi = \frac{5}{4} \pi \]

Substitute the radius \( r = 8 \) feet and the radian measure \( \theta = \frac{5}{4} \pi \) into the arc length formula: \[ \text{Arc length} = 8 \times \frac{5}{4} \pi \]

Simplify the expression to express the arc length in terms of \( \pi \), then multiply out to find the decimal approximation and round to two decimal places.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Arc Length Formula

The arc length of a circle is the distance along the curved line forming part of the circle's circumference. It is calculated using the formula s = rθ, where r is the radius and θ is the central angle in radians. This formula links linear distance to angular measure.

Conversion Between Degrees and Radians

Since the arc length formula requires the angle in radians, converting degrees to radians is essential. The conversion is done by multiplying degrees by π/180. For example, 225° equals (225 × π/180) radians, enabling correct use in trigonometric formulas.

Expressing Answers in Terms of π and Decimal Approximation

Often, answers are first expressed symbolically in terms of π to maintain exactness, then approximated numerically for practical use. After calculating arc length with π, rounding to two decimal places provides a usable decimal value, balancing precision and clarity.