A circle has a circumference of 12 pi feet (ft). An arc, x, in this circle has a central angle of 45^(@). What is the length of x ? Choose 1 answer: (A) (3pi)/(2)ft (B) 3pift (c) 270ft (D) 540ft

Convert to Fraction: To find the length of an arc (x) in a circle, we can use the formula arc length = (central angle/360) * circumference. The central angle is given as 45 degrees, and the circumference is given as 12 pi feet.Calculate Arc Length: First, we need to convert the central angle from degrees to a fraction of a full circle. Since a full circle is 360 degrees, a 45-degree angle is 45/360 of a full circle.Simplify Fraction: Now, we can calculate the arc length by multiplying the fraction of the circle that the angle represents by the total circumference. So, the arc length x = (45/360) * (12 pi).Multiply by Circumference: Simplify the fraction 45/360 by dividing both the numerator and the denominator by 45. This gives us 1/8.Simplify Expression: Now, multiply the simplified fraction by the circumference: x = (1/8) * (12 pi) = 12 pi / 8.Final Arc Length: Simplify the expression by dividing 12 by 8, which gives us 1.5 pi.The length of the arc x is therefore 1.5 pi feet, which can also be written as (3 pi) / 2 feet.

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Precalculus

Find trigonometric ratios using a Pythagorean or reciprocal identity

Full solution

Q. A circle has a circumference of

12 \pi

feet

(\mathrm{ft})

. An arc,

x

, in this circle has a central angle of

45^{\circ}

. What is the length of

x

\newline

Choose

1

answer:

\newline

(A)

\frac{3 \pi}{2} \mathrm{ft}

\newline

(B)

3 \pi \mathrm{ft}

\newline

(C)

270 \mathrm{ft}

\newline

(D)

540 \mathrm{ft}

Convert to Fraction: To find the length of an arc $x$ in a circle, we can use the formula arc length = $\frac{\text{central angle}}{360}$ * circumference. The central angle is given as $45$ degrees, and the circumference is given as $12 \pi$ feet.
Calculate Arc Length: First, we need to convert the central angle from degrees to a fraction of a full circle. Since a full circle is $360$ degrees, a $45$ -degree angle is $\frac{45}{360}$ of a full circle.
Simplify Fraction: Now, we can calculate the arc length by multiplying the fraction of the circle that the angle represents by the total circumference. So, the arc length $x = \left(\frac{45}{360}\right) \times (12 \pi)$ .
Multiply by Circumference: Simplify the fraction $\frac{45}{360}$ by dividing both the numerator and the denominator by $45$ . This gives us $\frac{1}{8}$ .
Simplify Expression: Now, multiply the simplified fraction by the circumference: $x = \frac{1}{8} \times (12 \pi) = \frac{12 \pi}{8}$ .
Final Arc Length: Simplify the expression by dividing $12$ by $8$ , which gives us $1.5 \pi$ .
Final Arc Length: Simplify the expression by dividing $12$ by $8$ , which gives us $1.5 \pi$ .The length of the arc $x$ is therefore $1.5 \pi$ feet, which can also be written as $(3 \pi) / 2$ feet.