Answer :
Answer:
To solve this problem, we'll start by finding the radius of the unit circle and then determine the scale factor needed to achieve the area of the new circle.
1. **Find the radius of the unit circle:**
The unit circle has a radius of 1.
2. **Area of the unit circle:**
The area of a circle is given by \( \pi r^2 \).
For the unit circle with \( r = 1 \):
\[
\text{Area} = \pi \cdot 1^2 = \pi
\]
3. **Find the radius of the new circle:**
We know the area of the new circle is \( 729\pi \) cm².
Using the area formula \( \pi R^2 = 729\pi \), we can solve for the new radius \( R \):
\[
R^2 = 729
\]
\[
R = \sqrt{729} = 27
\]
4. **Determine the scale factor:**
The scale factor \( k \) is the ratio of the new radius to the original radius.
\[
k = \frac{R}{r} = \frac{27}{1} = 27
\]
Therefore, the scale factor required to dilate the unit circle to a circle with an area of \( 729\pi \) cm² is 27.
