Question

The volume of a cone varies jointly with the area of the base and the height. when the area of the base is

Answer 1

The volume of a cone varies jointly with the area of the base and the height of the cone.

$V = kAh$

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

$V = \dfrac{1}{3}Ah$

Now we use the second set of info, the height and the volume, and we find the area of the base.

$V = \dfrac{1}{3}Ah$

$124 = \dfrac{1}{3} \times A \times 12$

$124 = 4A$

$A = 31$

Answer: the area of the base is 31 cm^2