Certainly! Let's begin by reviewing the concept of similar solids. When two geometric solids are similar, their corresponding linear dimensions are proportional.
In this case, we are given the dimensions of a right circular cone, specifically:
- Height = 3 feet
- Diameter of the base = 5 feet
From these dimensions, we can determine the radius of the base:
- Radius = Diameter / 2 = 5 feet / 2 = 2.5 feet
So, the right circular cone has the following dimensions:
- Height = 3 feet
- Radius = 2.5 feet
Given these measurements, if we want a similar cone, every corresponding dimension should increase or decrease proportionally according to a consistent scale factor.
Now, let's look at some possible new dimensions. Assume we have another right circular cone with the following dimensions to check for similarity:
1. Height = 9 feet
2. Height = 5 feet
3. Height = 10 feet
4. Height = 12 feet
5. Height = 25 feet
6. Height = 7 feet
7. Height = 6 feet
8. Height = 20 feet
### We can check the ratios of height to base radius for similarity:
#### Case 1:
- Height = 9 feet
- Scale factor for height [tex]\( = \frac{9}{3} = 3 \)[/tex]
- Radius should be [tex]\( = 2.5 \times 3 = 7.5 \)[/tex] feet (not given as diameter is unknown)
#### Case 2:
- Height = 5 feet
- Scale factor for height [tex]\( = \frac{5}{3} \approx 1.67 \)[/tex]
- Radius should be [tex]\( = 2.5 \times 1.67 \approx 4.18 \)[/tex] feet (not given as diameter is unknown)
#### Case 3:
- Height = 10 feet
- Scale factor for height [tex]\( = \frac{10}{3} \approx 3.33 \)[/tex]
- Radius should be [tex]\( = 2.5 \times 3.33 \approx 8.33 \)[/tex] feet (not given as diameter is unknown)
#### Case 4:
- Height = 12 feet
- Scale factor for height [tex]\( = \frac{12}{3} = 4 \)[/tex]
- Radius should be [tex]\( = 2.5 \times 4 = 10 \)[/tex] feet (not given, but assuming diameter of cone is 20)
#### Case 5:
- Height = 25 feet
- Scale factor for height [tex]\( = \frac{25}{3} \approx 8.33 \)[/tex]
- Radius should be [tex]\( = 2.5 \times 8.33 \approx 20.83 \)[/tex] feet (not given as diameter)
#### Case 6:
- Height = 7 feet
- Scale factor for height [tex]\( = \frac{7}{3} \approx 2.33 \)[/tex]
- Radius should be [tex]\( = 2.5 \times 2.33 \approx 5.83 \)[/tex] feet (not mentioned diameter)
#### Case 7:
- Height = 6 feet
- Scale factor for height [tex]\( = \frac{6}{3} = 2 \)[/tex]
- Radius should be [tex]\( = 2.5 \times 2 = 5 \)[/tex] feet (diameter should be 10)
#### Case 8:
- Height = 20 feet
- Scale factor for height [tex]\( = \frac{20}{3} \approx 6.67 \)[/tex]
- Radius should be [tex]\( = 2.5 \times 6.67 \approx 16.67 \)[/tex] feet (diameter should be 33)
All valid possible similar cones should have their diameter scaled accordingly by the same factor.
So the correct similar cone dimensions are directly in numerical relationship and proportional. The initial determination helps ensure we identify the cones that fit the definition of similar solids by proportions.
By understanding how each dimension scales proportionally, we can recognize the new dimensions.
