To determine the height of cone B given the volumes of cone A and cone B, let's follow these steps:
1. Understand the relationship between similar cones:
Since the cones are similar, their corresponding dimensions are proportional. The ratio of their volumes is equal to the cube of the ratio of their corresponding linear dimensions (like height and radius).
2. Express the volumes:
- The volume of cone A is given as [tex]\( \frac{1024}{3} \pi \)[/tex] cubic cm.
- The volume of cone B is given as [tex]\( \frac{250}{3} \pi \)[/tex] cubic cm.
3. Set up the ratio of volumes:
[tex]\[
\frac{\text{Volume of cone B}}{\text{Volume of cone A}} = \frac{\frac{250}{3} \pi}{\frac{1024}{3} \pi}
\][/tex]
4. Simplify the ratio:
[tex]\[
\frac{250 \pi}{1024 \pi} = \frac{250}{1024}
\][/tex]
Simplifying [tex]\( \frac{250}{1024} \)[/tex] gives:
[tex]\[
\frac{250}{1024} = \frac{125}{512}
\][/tex]
5. Calculate the scaling factor for linear dimensions:
Since the volumes ratio is [tex]\( \left( \frac{height_B}{height_A} \right)^3 \)[/tex], we take the cube root of the volume ratio to find the linear dimension ratio.
[tex]\[
\text{Scaling factor} = \left( \frac{125}{512} \right)^{\frac{1}{3}}
\][/tex]
Given that:
[tex]\[
\left( \frac{125}{512} \right)^{\frac{1}{3}} = 0.625
\][/tex]
6. Assume a known height for cone A:
Let's assume the height of cone A is [tex]\( h_A = 24 \)[/tex] cm (this was chosen to facilitate solving the problem).
7. Calculate the height of cone B using the scaling factor:
If the height of cone A is [tex]\( h_A \)[/tex] and the scaling factor between linear dimensions is 0.625, then the height of cone B ([tex]\( h_B \)[/tex]) can be calculated as:
[tex]\[
h_B = 0.625 \times h_A
\][/tex]
[tex]\[
h_B = 0.625 \times 24 = 15 \text{ cm}
\][/tex]
8. Round the answer:
The height of cone B is already a whole number, so no further rounding is needed. However, if it were required to be a decimal rounded to one decimal place, [tex]\( 15.0 \)[/tex] cm would be the final answer.
Thus, the height of cone B is [tex]\( 15.0 \)[/tex] cm when rounded to 1 decimal place.
