To find the radius [tex]\( r \)[/tex] of the two cones that make up the solid shape with a total surface area of [tex]\( 135.2 \pi \)[/tex] cm[tex]\(^2\)[/tex], we need to use the given information about the slant heights and the formula for the curved surface area of a cone. Here’s a step-by-step solution:
1. Identify the known values:
- Total surface area of the shape is [tex]\( 135.2 \pi \)[/tex] cm[tex]\(^2\)[/tex].
2. Set up the formulas for the curved surface areas of the cones:
For the first cone:
- Radius [tex]\( = r \)[/tex]
- Slant height [tex]\( l_1 = 2r \)[/tex]
- Curved surface area = [tex]\( \pi r l_1 = \pi r \cdot 2r = 2\pi r^2 \)[/tex]
For the second cone:
- Radius [tex]\( = r \)[/tex]
- Slant height [tex]\( l_2 = 3r \)[/tex]
- Curved surface area = [tex]\( \pi r l_2 = \pi r \cdot 3r = 3\pi r^2 \)[/tex]
3. Combine the curved surface areas to form one equation:
- Combined curved surface area = [tex]\( 2\pi r^2 + 3\pi r^2 = 5\pi r^2 \)[/tex]
4. Set this combined curved surface area equal to the total surface area:
[tex]\[
5\pi r^2 = 135.2 \pi
\][/tex]
5. Divide both sides of the equation by [tex]\(\pi\)[/tex]:
[tex]\[
5r^2 = 135.2
\][/tex]
6. Solve for [tex]\( r^2 \)[/tex]:
[tex]\[
r^2 = \frac{135.2}{5} = 27.04
\][/tex]
7. Solve for [tex]\( r \)[/tex]:
[tex]\[
r = \sqrt{27.04} = 5.2 \, \text{cm}
\][/tex]
So, the radius [tex]\( r \)[/tex] of each cone is [tex]\( 5.2 \)[/tex] cm.
