Sure! Let's go through the solution step-by-step.
1. Understanding the Problem
- We have two similar cones: a smaller cone with a surface area of 11.74 square inches.
- The scaling factor from the smaller cone to the larger cone is given as either [tex]\(\frac{3.5}{2.1}\)[/tex] or [tex]\(\frac{5}{3}\)[/tex].
2. Determine the Scale Factor
- First, verify the scale factor:
- [tex]\(\frac{3.5}{2.1} = \frac{35}{21} = \frac{5}{3}\)[/tex]
- This confirms that the scale factor from the smaller cone to the larger cone is indeed [tex]\(\frac{5}{3}\)[/tex].
3. Calculate the Square of the Scale Factor
- Since surface areas of similar figures change by the square of the scale factor:
[tex]\[
\left(\frac{5}{3}\right)^2 = \frac{5^2}{3^2} = \frac{25}{9}
\][/tex]
4. Set Up the Proportion
- Let [tex]\( x \)[/tex] be the surface area of the larger cone. From the given proportion:
[tex]\[
\frac{25}{9} = \frac{x}{11.74}
\][/tex]
5. Solve for [tex]\( x \)[/tex]
- To solve for [tex]\( x \)[/tex], we multiply both sides of the proportion by the surface area of the smaller cone (11.74):
[tex]\[
x = \frac{25}{9} \times 11.74
\][/tex]
- This gives us:
[tex]\[
x = 32.6111\ldots
\][/tex]
6. Round to the Nearest Hundredth
- Now round the result to the nearest hundredth:
[tex]\[
x \approx 32.61
\][/tex]
Therefore, the surface area of the larger cone is about [tex]\( 32.61 \)[/tex] square inches.
