find the exact length of the radius of a circle if the arc length cut off by an angle of 5pi/4 is 25pi/8. show all working 

We are given an arc length \( s = \frac{25\pi}{8} \) and a central angle \( \theta = \frac{5\pi}{4} \), and we need to find the radius \( r \) of the circle.

The formula for the arc length \( s \) of a circle is:

\[
s = r \theta
\]

Where:
- \( s \) is the arc length,
- \( r \) is the radius, and
- \( \theta \) is the angle in radians.

### Step-by-step solution:

1. **Substitute the known values** into the formula:

\[
\frac{25\pi}{8} = r \cdot \frac{5\pi}{4}
\]

2. **Solve for \( r \)**:
   - First, divide both sides by \( \frac{5\pi}{4} \):

\[
r = \frac{\frac{25\pi}{8}}{\frac{5\pi}{4}}
\]

3. **Simplify the right-hand side**:
   - Simplifying the division:

\[
r = \frac{25\pi}{8} \times \frac{4}{5\pi}
\]

4. **Cancel out \( \pi \) from the numerator and denominator**:

\[
r = \frac{25}{8} \times \frac{4}{5} = \frac{25 \times 4}{8 \times 5} = \frac{100}{40} = 2.5
\]

### Final answer:
The exact length of the radius is \( r = 2.5 \) units.

Would you like further details or explanations?

Here are five related questions you might find interesting:
1. What is the formula for the circumference of a circle?
2. How would the result change if the angle were given in degrees instead of radians?
3. What is the relationship between the radius and the diameter of a circle?
4. How do you find the area of a sector given the arc length and the radius?
5. How do you convert an angle from radians to degrees?

**Tip:** Always ensure angles are in radians when using formulas involving arc length in circular calculations.

Find the exact length of the radius of a circle if the arc length cut off by an angle of 5pi/4 is 25pi/8. Show all working.

Learn how to find the exact length of the radius of a circle when given the arc length and central angle in radians. This detailed step-by-step solution uses the arc length formula and simplifies expressions to solve for the radius. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation