Answer:
[tex]\frac{7}{8}[/tex]
Step-by-step explanation:
In any circle the following ratios are equal
[tex]\frac{centralangle}{2\pi }[/tex] = [tex]\frac{arc}{circumference}[/tex]
[tex]\frac{\frac{7\pi }{4} }{2\pi }[/tex] = [tex]\frac{7}{8}[/tex] = [tex]\frac{arc}{circumference}[/tex]
The arc is 7/8 of the circumference.
From the question, an arc subtends a central angle measuring 7pi/4 radians.
Let the central angle of the arc be θ
∴ [tex]\theta = \frac{7\pi}{4} rad[/tex]
To determine the fraction of the circumference the arc is, that is, the ratio of the length of the arc to the measure of circumference.
Length of an arc is given by the formula
[tex]s=r\theta\\[/tex]
where s is the arc length, r is the radius and θ is the central angle in radians
∴ The length of the arc = [tex]\frac{7\pi}{4} \times r[/tex]
Length of the arc = [tex]\frac{7\pi r}{4}[/tex]
Now, for the circumference of the circle.
Circumference of a circle is given by the formula
[tex]C = 2\pi r[/tex]
∴ The fraction of the circumference which the arc is, is
[tex]\frac{7\pi r}{4} \div2\pi r[/tex]
= [tex]\frac{7\pi r}{4} \times \frac{1}{2\pi r}[/tex]
= [tex]\frac{7\pi r}{8\pi r}[/tex]
[tex]= \frac{7}{8}[/tex]
Hence, the arc is 7/8 of the circumference.
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