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Difficulty: HardVolume and Surface Area of Solids

A right circular cylindrical tank with a base radius of 4 inches4\text{ inches} contains water. A solid metal sphere with a radius of 3 inches3\text{ inches} is completely submerged in the water, causing the water level to rise. If no water overflows from the tank, what is the increase, in inches, in the water level of the tank?

  1. A
    0.5625
  2. B
    9.00
  3. 2.25Answer
  4. D
    6.75

Answer

2.25
The correct answer is 2.25. The volume of the submerged sphere is calculated using the formula V=43πr3V = \frac{4}{3}\pi r^3. With a radius of 3 inches3\text{ inches}, the volume is V=43π(3)3=36π cubic inchesV = \frac{4}{3}\pi (3)^3 = 36\pi\text{ cubic inches}. The volume of the displaced water is represented by a cylinder of radius 4 inches4\text{ inches} and height hh, which represents the increase in water level. The volume of this displaced cylinder is V=π(4)2h=16πhV = \pi (4)^2 h = 16\pi h. Since the volume of the displaced water is equal to the volume of the submerged sphere, we set them equal: 16πh=36π16\pi h = 36\pi. Dividing both sides by 16π16\pi yields h=2.25h = 2.25.

Step-by-Step Solution

1
Calculate the volume of the solid metal sphere using the formula V=43πr3V = \frac{4}{3}\pi r^3 with radius r=3 inchesr = 3\text{ inches}.
Vsphere=43π(3)3=36π cubic inchesV_{\text{sphere}} = \frac{4}{3}\pi (3)^3 = 36\pi\text{ cubic inches}
The volume of the sphere represents the total volume of water that will be displaced when the sphere is completely submerged.
2
Express the volume of the displaced water in the cylinder as a function of the rise in water level, hh.
Vdisplaced=π(4)2h=16πh cubic inchesV_{\text{displaced}} = \pi (4)^2 h = 16\pi h\text{ cubic inches}
The displaced water forms a cylindrical shape with the same base radius as the tank (4 inches4\text{ inches}) and a height equal to the change in the water level (hh).
3
Equate the volume of the displaced water to the volume of the sphere and solve for hh.
16πh=36π⇒h=3616=2.25 inches16\pi h = 36\pi \Rightarrow h = \frac{36}{16} = 2.25\text{ inches}
By Archimedes' principle, the volume of the displaced water must equal the volume of the fully submerged solid sphere.

Key Concept

Equating the volume of a submerged object to the volume of displaced liquid in a cylindrical container.
Estimated Time:2m 0s
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Topics

Volume and Surface Area of Solids

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