To find the radius of the circle drawn through the vertices of the cones, consider the arrangement of the cones. Each cone is touching the other two at their bases, forming an equilateral triangle when viewed from above.
1. Arrangement and Geometry:
- The centers of the three cones' bases form an equilateral triangle with side length \(2r\) (since the distance between the centers of two touching cones is twice the radius, \(r\)).
- The vertices of the cones are directly above these centers.
2. Circumcircle of the Equilateral Triangle:
- The radius \(R\) of the circumcircle of an equilateral triangle with side length \(2r\) can be found using the formula \(R = \frac{a}{\sqrt{3}}\), where \(a\) is the side length of the triangle.
3. Calculation:
- For our equilateral triangle with side length \(2r\), the circumradius \(R\) is:
\[ R = \frac{2r}{\sqrt{3}} = \frac{2r \sqrt{3}}{3}\]
4. Comparison:
- Simplifying, \( \frac{2r \sqrt{3}}{3} \) is clearly larger than \(r\), because \( \sqrt{3} \) is approximately 1.732, making \( \frac{2r \times 1.732}{3} \approx 1.154r \).
Thus, the radius of the circle drawn through the vertices of the cones is larger than \(r\).
Explanation: Since the radius of the circumcircle of the equilateral triangle formed by the centers of the cones' bases is larger than the radius of the cones' bases, the radius of the circle drawn through the vertices of the cones will be larger than \(r\).
Therefore, the correct answer is C Larger than r.
How many pairs of letters are there in the word 'LANGUISH' which have the same letters between them in the word as in the alphabet?
Consider the following alphanumeric series with powers:
A1, C3, E5, G7, __, __, I9, __,K11, M13, __
Based on the observed pattern, complete the series by selecting the correct options:
Given the statements:
1. All smartphones are devices.
2. Some devices are expensive.
Conclusions:
I. Some expensive things are smartphones.
II. All smartphones are expensive. Select the correct conclusions:
Consider the following information:
Set A: Animals that can fly
Set B: Birds
Set C: Animals that live in water
Using Venn diagrams, represent the relationships between these sets and answer the question. Which region(s) in the Venn diagram represents animals that can fly and also live in water?
Arrange the following words in lexicographical (dictionary) order from highest to lowest:
1. Elephant
2. Banana
3. Apple
4. Cherry
A trader marked up shirts by 40%, offered a 20% discount during a sale, and sold each for 234. Find the number of shirts he purchased.