Respuesta :
Answer:
18 cm
Step-by-step explanation:
If we take the cross section of this cone through both the centers of these circles, we can create a 2d diagram which will be easier to interpret (I will be referring to the diagram that I have attached below). By the definition of a frustum, EF must be parallel to BC because a frustum is cut by a plane parallel to the original cone's base. Hence, we have corresponding angles with angle AEF and ABC. Since, triangles AEF and ABC share angle ABC, we know that triangles AEF and ABC are similar by Angle-Angle similarity. We also know the side ratios of these to be 1:3 as the bottom diameter (BC) is thrice the top diameter (EF). This means that the heights of triangles AEF and ABC will also be in a ratio of 1:3.
Let the height of triangle AEF be [tex]x[/tex]. If [tex]\frac{\text{height of AEF}}{\text{height of ABC}} = \frac13[/tex], then we can substitute this with their values in terms of [tex]x[/tex]. Now, we need to solve the equation:
[tex]\frac{x}{x+12} = \frac13[/tex].
Cross multiplying gives us
[tex]3x = x + 12[/tex]
We can subtract by x to get
[tex]2x=12[/tex]
Dividing by 2 gives us [tex]x=6[/tex]
If the height of AEF is 6, then we know that the height of the whole cone (the height of ABC) will be 6+12=18. Therefore, the height of the whole cone is [tex]\boxed{18 \text{ cm}}.[/tex]
