Question

type the correct answer in each box. use numerals instead of words. consider the given circle with the shaded sector xy and central angle, 225°. the circumference of the circle is 30π units. use the given information to complete the statements. round any non-integer answers to the hundredths place. the length of major arc xy is units. the radius of the circle is units. the area of the shaded sector is square units.

Explanation:

Step1: Find the radius of the circle

The formula for the circumference of a circle is \( C = 2\pi r \), where \( C = 30\pi \).
Set \( 2\pi r = 30\pi \), divide both sides by \( 2\pi \): \( r=\frac{30\pi}{2\pi}=15 \).

Step2: Find the length of major arc XY

The central angle of the major arc is \( 225^\circ \). The formula for arc length is \( L=\frac{\theta}{360^\circ}\times C \), where \( \theta = 225^\circ \) and \( C = 30\pi \).
Substitute values: \( L=\frac{225^\circ}{360^\circ}\times30\pi=\frac{5}{8}\times30\pi=\frac{150\pi}{8}=18.75\pi\approx47.12 \) (wait, no, wait: Wait, major arc? Wait, the central angle for the shaded sector is \( 225^\circ \), so the arc length of the shaded (major) arc: Wait, the circumference is \( 30\pi \), so arc length is \( \frac{225}{360}\times30\pi \). Let's calculate: \( \frac{225}{360}=\frac{5}{8} \), so \( \frac{5}{8}\times30\pi=\frac{150\pi}{8}=18.75\pi\approx47.12 \)? Wait, no, wait, \( 30\pi \) is circumference. Wait, \( 225/360 = 5/8 \), so \( 5/8 * 30\pi = (150/8)\pi = 18.75\pi \approx 58.90 \)? Wait, no, \( \pi \approx 3.1416 \), so \( 18.75 * 3.1416 \approx 58.90 \). Wait, maybe I made a mistake. Wait, let's recalculate:

Wait, circumference \( C = 30\pi \), so the arc length formula is \( L = \frac{\theta}{360^\circ} \times C \). For \( \theta = 225^\circ \), \( L = \frac{225}{360} \times 30\pi \). Simplify \( 225/360 = 5/8 \), so \( 5/8 * 30\pi = (150/8)\pi = 18.75\pi \approx 18.75 * 3.14159265 \approx 58.90 \).

Step3: Find the area of the shaded sector

The formula for the area of a sector is \( A=\frac{\theta}{360^\circ}\times\pi r^2 \), where \( \theta = 225^\circ \) and \( r = 15 \).
Substitute values: \( A=\frac{225^\circ}{360^\circ}\times\pi\times15^2=\frac{5}{8}\times\pi\times225=\frac{1125\pi}{8}\approx\frac{1125\times3.1416}{8}\approx\frac{3534.3}{8}\approx441.79 \). Wait, no: \( 15^2 = 225 \), \( 225\times\pi = 225\pi \), \( 225/360 = 5/8 \), so \( 5/8 * 225\pi = (1125/8)\pi \approx 140.625\pi \approx 441.79 \). Wait, but let's check the radius first. Wait, radius: circumference \( C = 2\pi r = 30\pi \), so \( r = 15 \), that's correct.

Wait, let's re-express:

1. Radius: \( C = 2\pi r \implies 30\pi = 2\pi r \implies r = 15 \). Correct.

2. Length of major arc XY: The central angle is \( 225^\circ \), so arc length \( L = \frac{225}{360} \times C = \frac{5}{8} \times 30\pi = \frac{150\pi}{8} = 18.75\pi \approx 58.90 \) (since \( \pi \approx 3.1416 \), \( 18.75 \times 3.1416 = 58.905 \approx 58.91 \)? Wait, 18.75 * 3.1416: 18 * 3.1416 = 56.5488, 0.75 * 3.1416 = 2.3562, total 56.5488 + 2.3562 = 58.905, so 58.91 when rounded to hundredths.

3. Area of shaded sector: \( A = \frac{225}{360} \times \pi r^2 = \frac{5}{8} \times \pi \times 15^2 = \frac{5}{8} \times 225\pi = \frac{1125\pi}{8} \approx \frac{1125 \times 3.1416}{8} \approx \frac{3534.3}{8} \approx 441.79 \) (since 3534.3 / 8 = 441.7875 ≈ 441.79).

Wait, but let's check the arc length again. Wait, the problem says "major arc XY". The central angle for the major arc: since the shaded sector has central angle 225°, which is more than 180°, so that's the major arc. So the arc length is indeed \( \frac{225}{360} \times 30\pi \). Let's compute that: 225/360 = 0.625, 0.625 * 30π = 18.75π ≈ 58.91 (since π ≈ 3.1416, 18.75 * 3.1416 = 58.905 ≈ 58.91).

Radius: 15, correct.

Area: \( \frac{225}{360} \times \pi \times 15^2 = \frac{5}{8} \times 225\pi = \frac{1125\pi}{8} \approx 441.79 \).

So:

- Length of major arc XY: \( 18.75\pi \approx 58.91 \) (wait, but let's check the circumference is 30π, so 30π is about 94.25. Then 225/360 of that is (225/360)*94.25 ≈ 0.625*94.25 ≈ 58.91. Correct.

- Radius: 15.

- Area: (225/360)*π*15² = (5/8)*225π = 1125π/8 ≈ 441.79.

Answer:

The length of major arc XY is \(\boldsymbol{58.91}\) units. The radius of the circle is \(\boldsymbol{15}\) units. The area of the shaded sector is \(\boldsymbol{441.79}\) square units.