Question

The inside of a beaker is shaped like a circular cylinder with a base diameter of 121212 centimeters (\text{cm})(cm)(, start text, c, m, end text, ). Water in the beaker is filled to a height of 8\,\text{cm}8cm8, start text, c, m, end text above the base. A stainless steel sphere with a diameter of 6\,\text{cm}6cm6, start text, c, m, end text submerged in the liquid displaces an equal volume of water. How high above the base of the beaker is the new water level, in centimeters

Answer 1

9 cm

Explanation:

Since the container is a cylinder of base diameter d₁ = 12 cm and thus radius r = 12/2 = 6 cm. The water height is 8 cm. The initial volume of water is thus V₁ = πr²h = π × 6² × 8 = 288π cm³.

The sphere of diameter d = 6 cm has radius r₁ = 6/2 = 3 cm. Its volume V₂ = 4πr³/3 = 4π × 3³/3 = 36π cm³

Since the sphere displaces it own volume of liquid, the new volume V = V₁ + V₂ = 288π cm³ + 36π cm³ = 324π cm³

This is the new volume of water in the cylinder. Since the water level rises by h' cm and V = πr²h'

h' = V/πr²

= 324π cm³/π(6)²

= 324π cm³/36π cm²

= 9 cm

So, the water level is 9 cm above the base