jmc

Question

A 5 8 rectangle can be rolled to form two different cylinders with different maximum volumes. What is the ratio of the larger volume to the smaller volume? Express your answer as a common fraction.

Accepted Answer

Holding the rectangle vertically, we can form a cylinder with height 8 and base circumference of 5. Let this cylinder have volume

V_{A}

and radius

r_{A}

; we have

2 π r_{A} = 5

so

r_{A} = \frac{5}{2 π}

and

V_{A} = π r_{A}^{2} h = π (\frac{5}{2 π})^{2} (8) = \frac{50}{π}

.

Holding the rectangle horizontally, we can form a cylinder with height 5 and base circumference of 8. Similarly, let this cylinder have volume

V_{B}

and radius

r_{B}

; we have

2 π r_{B} = 8

so

r_{B} = \frac{4}{π}

and

V_{B} = π r_{B}^{2} h = π (\frac{4}{π})^{2} (5) = \frac{80}{π}

.

Hence, the ratio of the larger volume to the smaller volume is

\frac{80/ π}{50/ π} = \frac{8}{5}

.