A cylindrical glass with an inside radius of 4 centimeters is... | SAT Area and volume Question, Answer Explained

Math · Geometry and TrigonometryHard

A cylindrical glass with an inside radius of $4$ centimeters is partially filled with water. A solid metal ball with a radius of $3$ centimeters is placed in the glass and sinks until it is completely under water. No water spills out. By how many centimeters does the water level rise?

$\frac{27}{16}$

$\frac{9}{4}$

$\frac{9}{2}$

$9$

Show answer and explanation

Answer: B

The water rises by a volume equal to the volume of the ball, $\frac{4}{3}\pi(3)^3 = 36\pi$ cubic centimeters. This volume fills a cylinder of radius $4$ and height $h$ , so $\pi(4)^2 h = 36\pi$ . Then $16h = 36$ , and $h = \frac{9}{4}$ centimeters.

Choice A leaves the $\frac{4}{3}$ out of the volume of the ball, using $\pi(3)^3 = 27\pi$ instead of $36\pi$ .

Why the other choices are wrong

A. This uses π times 3 cubed for the ball and leaves out the 4/3.
C. This divides by the circumference of the glass, 8π, instead of the area of its base, 16π.
D. This divides by 4π instead of 16π, so the radius of the glass wasn't squared.

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