Given two right cones, one with a base radius of $4$ units and height $x$ units, and another with a base radius of $8$ units and height $12$ units, which value of $x$ would make the two cones similar?

• A. $8$
• B. $6$
• C. $4$
• D. $12$

Answer: (B)

Verified step by step guidance

1. Recall that two cones are similar if their corresponding linear dimensions are proportional. This means the ratio of their base radii must be equal to the ratio of their heights.
2. Set up the proportion using the given radii and heights: \(\frac{4}{8} = \frac{x}{12}\), where 4 and 8 are the base radii of the smaller and larger cones respectively, and \(x\) and 12 are their heights.
3. Simplify the ratio of the radii: \(\frac{4}{8} = \frac{1}{2}\), so the proportion becomes \(\frac{1}{2} = \frac{x}{12}\).
4. Solve for \(x\) by cross-multiplying: \$1 \times 12 = 2 \times x\(, which simplifies to \)12 = 2x$.
5. Divide both sides by 2 to isolate \(x\): \(x = \frac{12}{2}\), which gives the value of \(x\) that makes the two cones similar.