Answer:
a) 1/4
b) 64:1
Step-by-step explanation:
a) The ratio of their heights:
Since the cylinders are similar, the ratio of their radii is equal to the ratio of their heights. That is, r1/r2 = h1/h2.
Given that the radii of the cylinders are 4 cm and 16 cm, we get r1/r2 = 4/16 = 1/4.
Therefore, the ratio of their heights h1/h2 = r1/r2 = 1/4
b) The ratio of their volumes:
The volume of a cylinder is given by the formula V = πr²h. Because the cylinders are similar, we can use the ratio of their radii and heights to find the ratio of their volumes.
Let V1 and V2 be the volumes of the smaller and larger cylinders.
V2/V1 = (π * (16 cm)² * h2) / (π * (4 cm)² * h1)
Because the cylinders are similar, h2/h1 = 16 cm / 4 cm = 4 (from part a).
Substituting h2/h1 into the equation gives:
V2/V1 = (π * (16 cm)² * 4) / (π * (4 cm)² * 1)
Simplify by canceling out similar terms:
V2/V1 = (16 cm / 4 cm)³ = 4³ = 64
So, the ratio of the volumes of the two cylinders is 64:1.
:)