jmc

Question

Amy and Belinda each roll a sheet of 6-inch by 8-inch paper to form a cylindrical tube. Amy tapes the two 8-inch sides together without overlap. Belinda tapes the two 6-inch sides together without overlap. What is times the positive difference of the volumes of the two tubes?

Accepted Answer

Amy's cylinder has a height of 8 and a base circumference of 6. Let her cylinder have volume

V_{A}

and radius

r_{A}

; we have

2 π r_{A} = 6

so

r_{A} = 3/ π

and

V_{A} = π r_{A}^{2} h = π (3/ π)^{2} (8) = 72/ π

.

Belinda's cylinder has a height of 6 and a base circumference of 8. Similarly, let her cylinder have volume

V_{B}

and radius

r_{B}

; we have

2 π r_{B} = 8

so

r_{B} = 4/ π

and

V_{B} = π r_{B}^{2} h = π (4/ π)^{2} (6) = 96/ π

.

The positive difference between the volume of the two tubes is

96/ π - 72/ π = 24/ π

cubic inches;

π

times this difference is

24

cubic inches.