Problem: The radius R​ of a cylindrical box is 8 inches, the height H, is 3 inches. The volume V=πR2H is to be increased by the same fixed positive amount when R​ is increased by x​ inches as when H​ is increased by x​ inches. This condition is satisfied by:
Answer Choices:
A. no real value of x
B. one integral value of x
C. one rational, but not integral, value of x
D. one irrational value of x
E. two real values of x
Solution:
V=Ï€R2H,Ï€(R+x)2H=Ï€R2(H+x)
R2H+2RxH+x2H=R2H+R2x
2Rxh+x2H=R2x
since xî€ =02RH+xH=R2,x=HP2−2RH​
For R=8,H=3x=316​, so that (C) is the correct choice.