baker2145 wrote:
Here's the question:
A 10-by-6 inch piece of paper is used to form the lateral surface of a cylinder. If the entire piece of paper is used to make the lateral surface, which of the following must be true of the two possible cylinders that can be formed?
A The volume of the cylinder with height 10 is 60/pi cubic inches greater than the volume of the cylinder with height 6.
B The volume of the cylinder with height 6 is 60/pi cubic inches greater than the volume of the cylinder with height 10.
C The volume of the cylinder with height 10 is 60pi cubic inches greater than the volume of the cylinder with height 6.
D The volume of the cylinder with height 6 is 60pi cubic inches greater than the volume of the cylinder with height 10.
E The volume of the cylinder with height 6 is 240/pi cubic inches greater than the volume of the cylinder with height 10.
I am having difficulty with this one: Please help and explain, if answer if B or E, WHY it is (#/pi) and not #pi, considering pi(r-squared)(h) is the volume formula
Thanks,
Hopefully, you have some intuition about which of the possible cylinders is going to have the greater volume. Because the volume of a cylinder is directly proportional to the height and directly proportional to the square of the radius, the size of the radius has the greatest effect on the volume of the cylinder. Given this intuition we can eliminate A and C.
The answer given already is correct, but I'm going to provide some more details in term of calculation
Cylinder with height 10 and circumference 6:
\(2pi(r) = 6\)
\(r = \frac{3}{(pi)}\)
\(V = (pi)(\frac{3}{pi})^2(10)\)
\(V = \frac{90}{pi}\)
Cylinder with height 6 and circumference 10:
\(2pi(r) = 10\)
\(r = \frac{5}{(pi)}\)
\(V = (pi)(\frac{5}{pi})^2(6)\)
\(V = \frac{150}{pi}\)
150-90 = 60
So, B.