A sector has an area of `40pi` and a radius of 10. What is the arc length of the sector?

Answer

To find the arc length of a sector given its area and radius, we can follow these steps:

### Step 1: Write down the formula for the area of a sector.
The area \( A \) of a sector is given by the formula:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
where \( \theta \) is the central angle in degrees and \( r \) is the radius.

### Step 2: Substitute the given values into the area formula.
We know that the area \( A = 40\pi \) and the radius \( r = 10 \). Substituting these values into the formula gives:
\[ 40\pi = \frac{\theta}{360} \times \pi \times 10^2 \]
This simplifies to:
\[ 40\pi = \frac{\theta}{360} \times \pi \times 100 \]

### Step 3: Simplify the equation.
We can cancel \( \pi \) from both sides:
\[ 40 = \frac{\theta}{360} \times 100 \]
Now, multiply both sides by \( 360 \):
\[ 40 \times 360 = \theta \times 100 \]
This simplifies to:
\[ 14400 = \theta \times 100 \]

### Step 4: Solve for \( \theta \).
Dividing both sides by 100 gives:
\[ \theta = \frac{14400}{100} = 144 \text{ degrees} \]

### Step 5: Convert \( \theta \) from degrees to radians.
To convert degrees to radians, use the conversion factor \( \frac{\pi}{180} \):
\[ \theta \text{ (in radians)} = 144 \times \frac{\pi}{180} \]
Simplifying this:
\[ \theta = \frac{144\pi}{180} = \frac{4\pi}{5} \text{ radians} \]

### Step 6: Use the arc length formula.
The arc length \( L \) of a sector is given by:
\[ L = r \times \theta \]
Substituting \( r = 10 \) and \( \theta = \frac{4\pi}{5} \):
\[ L = 10 \times \frac{4\pi}{5} \]

### Step 7: Simplify to find \( L \).
Calculating this gives:
\[ L = \frac{40\pi}{5} = 8\pi \]

### Final Answer:
The arc length of the sector is \( 8\pi \).

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