Two congruent cylinders each have radius 8 inches and height 3 inches. The radius of one cylinder and the height of the other are both increased by the same number of inches. The resulting volumes are equal. How many inches is the increase? Express your answer as a common fraction.
V=pi x r^2 x h
V=pi x 8^2 x 3
V=192 in^3 before increase.
Let the increase =n
Pi x (8+n)^2 x 3=Pi x 8^2 x (3+n), solve for n
3 π (n + 8)^2 = 64 π (n + 3)
Divide both sides by π:
3 (n + 8)^2 = 64 (n + 3)
Expand out terms of the right hand side:
3 (n + 8)^2 = 64 n + 192
Subtract 64 n + 192 from both sides:
-192 - 64 n + 3 (n + 8)^2 = 0
Expand out terms of the left hand side:
3 n^2 - 16 n = 0
Factor n from the left hand side:
n (3 n - 16) = 0
Split into two equations:
n = 0 or 3 n - 16 = 0
Add 16 to both sides:
n = 0 or 3 n = 16
Divide both sides by 3:
n = 16/3 =5 1/3 inches
