Question

1. a circle has a circumference whose length is $25\pi$. find the length of an arc whose central angle is $90^\circ$. ad. is 11.634

Explanation:

Step1: Recall arc length formula

The length of an arc \( s \) is given by the formula \( s=\frac{\theta}{360^{\circ}}\times C \), where \( \theta \) is the central angle in degrees and \( C \) is the circumference of the circle.

Step2: Substitute the given values

We know that \( \theta = 90^{\circ} \) and \( C = 25\pi \). Substituting these values into the formula, we get \( s=\frac{90^{\circ}}{360^{\circ}}\times25\pi \).

Step3: Simplify the expression

Simplify \( \frac{90^{\circ}}{360^{\circ}}=\frac{1}{4} \). Then \( s = \frac{1}{4}\times25\pi=\frac{25\pi}{4}= 6.25\pi\approx6.25\times3.1416 = 19.635 \) (if we approximate \( \pi\approx3.1416 \)). But if we keep it in terms of \( \pi \), it is \( \frac{25\pi}{4} \), and as a decimal (using \( \pi\approx3.14 \)): \( \frac{25\times3.14}{4}=\frac{78.5}{4} = 19.625\approx19.63 \) (close to the given \( 19.634 \)).

Answer:

The length of the arc is \( \frac{25\pi}{4} \) (or approximately \( 19.63 \))