A solid right circular cylinder has a radius of r centimeters and a height of 6 centimeter… — SAT

A solid right circular cylinder has a radius of $r$ centimeters and a height of $6$ centimeters. A solid sphere has a radius of $r$ centimeters. If the sum of the volumes of the cylinder and the sphere is equal to the volume of a right circular cylinder with a radius of $2r$ centimeters and a height of $2$ centimeters, what is the value of $r$ ?

Answer: 1.5 centimeters

Answer

The correct answer is 1.5 (or 3/2).

The correct answer is 1.5. By using the formulas for the volume of a cylinder (

V = \pi r^2 h

) and a sphere (

V = \frac{4}{3}\pi r^3

), the total volume of the first cylinder and the sphere is

6\pi r^2 + \frac{4}{3}\pi r^3

. The second cylinder has a volume of

\pi(2r)^2(2) = 8\pi r^2

. Setting them equal gives

6\pi r^2 + \frac{4}{3}\pi r^3 = 8\pi r^2

. Subtracting

6\pi r^2

from both sides gives

\frac{4}{3}\pi r^3 = 2\pi r^2

. Dividing both sides by

\pi r^2

(since

r > 0

) simplifies to

\frac{4}{3}r = 2

, which solves to

r = 1.5

Step-by-Step Solution

Express the volume of the first cylinder using the formula

V = \pi r^2 h

V_1 = 6\pi r^2

To represent the cylinder's volume in terms of its radius.

Express the volume of the sphere using the formula

V = \frac{4}{3}\pi r^3

V_2 = \frac{4}{3}\pi r^3

To represent the sphere's volume in terms of its radius.

Express the volume of the second cylinder, which has a radius of

2r

and a height of

2

V_3 = \pi (2r)^2(2) = 8\pi r^2

To represent the larger cylinder's volume in terms of

r

Set up the equation where the sum of the first cylinder and sphere volumes equals the second cylinder volume.

6\pi r^2 + \frac{4}{3}\pi r^3 = 8\pi r^2

To relate the volumes as given in the problem statement.

Simplify the equation by subtracting

6\pi r^2

from both sides.

\frac{4}{3}\pi r^3 = 2\pi r^2

To isolate the cubic term on one side.

Divide both sides of the equation by

\pi r^2

since

r > 0

\frac{4}{3}r = 2

To reduce the polynomial equation to a linear equation.

Multiply both sides by

\frac{3}{4}

to solve for

r

r = 1.5

To find the final numerical value of the radius.

Key Concept

Volume of cylinders and spheres

Question

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Step-by-Step Solution

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