Question

question 6 (1 point)
a circular ride has a radius of 12 m. one section of the ride covers a 30° angle.
what is the area of that section?
______ m²
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question 7 (1 point)
a dartboard has a radius of 9 cm. one scoring region is a 40° sector.
find the area of that region.
______ cm²
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question 8 (1 point)
a circular stained - glass window has a radius of 4 ft. one decorative panel forms a 135° sector.
what is the area of the panel?
______ ft²
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Explanation:

Question 6

Step 1: Recall the formula for the area of a sector

The area of a sector of a circle is given by \( A = \frac{\theta}{360^\circ} \times \pi r^2 \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.

Step 2: Identify the values of \( \theta \) and \( r \)

Here, \( \theta = 30^\circ \) and \( r = 12 \) m.

Step 3: Substitute the values into the formula

First, calculate \( r^2 \): \( 12^2 = 144 \).
Then, calculate the fraction \( \frac{30^\circ}{360^\circ} = \frac{1}{12} \).
Now, multiply by \( \pi r^2 \): \( A = \frac{1}{12} \times \pi \times 144 \).
Simplify: \( \frac{144}{12} \pi = 12\pi \approx 37.7 \) (if we use \( \pi \approx 3.14 \), \( 12\times3.14 = 37.68 \)).

Step 1: Recall the sector area formula

The formula for the area of a sector is \( A = \frac{\theta}{360^\circ} \times \pi r^2 \), where \( \theta \) is the central angle and \( r \) is the radius.

Step 2: Identify \( \theta \) and \( r \)

Here, \( \theta = 40^\circ \) and \( r = 9 \) cm.

Step 3: Substitute into the formula

First, find \( r^2 \): \( 9^2 = 81 \).
Then, the fraction \( \frac{40^\circ}{360^\circ} = \frac{1}{9} \).
Multiply by \( \pi r^2 \): \( A = \frac{1}{9} \times \pi \times 81 \).
Simplify: \( \frac{81}{9} \pi = 9\pi \approx 28.3 \) (using \( \pi \approx 3.14 \), \( 9\times3.14 = 28.26 \)).

Step 1: Recall the sector area formula

The area of a sector is \( A = \frac{\theta}{360^\circ} \times \pi r^2 \), with \( \theta \) as the central angle and \( r \) as the radius.

Step 2: Identify \( \theta \) and \( r \)

Here, \( \theta = 135^\circ \) and \( r = 4 \) ft.

Step 3: Substitute into the formula

First, calculate \( r^2 \): \( 4^2 = 16 \).
Then, the fraction \( \frac{135^\circ}{360^\circ} = \frac{3}{8} \).
Multiply by \( \pi r^2 \): \( A = \frac{3}{8} \times \pi \times 16 \).
Simplify: \( \frac{3\times16}{8} \pi = 6\pi \approx 18.8 \) (using \( \pi \approx 3.14 \), \( 6\times3.14 = 18.84 \)).

Answer:

\( 12\pi \) (or approximately \( 37.7 \))

Question 7