The volume of a cone varies jointly with the area of the base and the height of the cone.
[tex] V = kAh [/tex]
where
V = volume
k = constant of proportionality
A = area of the base
h = height of the cone
The given info is for A = 27, and h = 6, then V = 54 (all using appropriate units for area, length, and volume, respectively.)
Now we can find k, the constant of proportionality.
V = kAh
54 = k * 27 * 6
k = 54/(27 * 6)
k = 1/3
The formula is
[tex] V = \dfrac{1}{3}Ah [/tex]
Now we use the second set of info, the height and the volume, and we find the area of the base.
[tex] V = \dfrac{1}{3}Ah [/tex]
[tex] 124 = \dfrac{1}{3} \times A \times 12 [/tex]
[tex] 124 = 4A [/tex]
[tex] A = 31 [/tex]
Answer: the area of the base is 31 cm^2
