cylinder

Cylinders X and Y are right cylinders with equal volumes.

Cylinder X has a height of 5 inches and a radius of 6 inches.

Cylinder Y has a height of 9 inches.

What is the radius of Cylinder Y, in inches

Radius of Cylinder X \(= r_x\)
Height of Cylinder X \(= h_x\)

Radius of Cylinder Y \(= r_y\)
Height of Cylinder Y \(= h_y\)

\(\begin{array}{rclcll} V &=& \pi \cdot r_y^2 \cdot h_y &=& \pi \cdot r_x^2 \cdot h_x \\ && \not{\pi} \cdot r_y^2 \cdot h_y &=& \not{\pi} \cdot r_x^2 \cdot h_x \\ && r_y^2 \cdot h_y &=& r_x^2 \cdot h_x \qquad & | \qquad : h_y\\ && r_y^2 &=& r_x^2 \cdot \frac{ h_x } { h_y } \qquad & | \qquad \sqrt{}\\ && r_y &=& r_x \cdot \sqrt{ \frac{ h_x } { h_y } } \qquad & | \qquad h_x = 5\ in. \qquad h_y = 9\ in. \qquad r_x = 6\ in.\\ && r_y &=& 6\ in. \cdot \sqrt{ \frac{ 5\ in. } { 9\ in. } } \\ && r_y &=& 6 \cdot \sqrt{ \frac{ 5 } { 9 } } \ in.\\ && r_y &=& 6 \cdot \frac{ \sqrt{ 5 } } { 3 } \ in.\\ && r_y &=& 2 \cdot \sqrt{ 5 } \ in.\\ && \mathbf{r_y} &\mathbf{=}& \mathbf{4.472135955 \ in.} \end{array}\)

The formula for the volume of a cylinder is:  V  =  pi · r² · h     (r = radius; h = height)

Cylinder X:  V  =  pi · 6² · 5     --->   V  =  180pi

Cylinder Y;  V  =  pi · r² · 9     --->   V  =  9pi · r²

Since the volumes are equal:   180pi  =  9pi · r²

Divide both sides by 9pi:                 20  =  r²

Find the square root:                         r  =  sqrt(20)  =  4.47 inches  (approximately)