Question

The radius $R$ of a cylindrical box is $8$ cm, the height $H$ is $3$ cm. The volume $V=\pi R^{2}H$ is to be increased by the same fixed positive amount when $R$ is increased by $x$ inches as when $H$ is increased by $x$ cm. This condition is satisfied by

• A. No real value of $x$
• B. One integer value of $x$
• C. One rational, but not integral, value of $x$
• D. One irrational value of $x$
• E. Two real values of $x$

Answer 1

The correct option is C One rational, but not integral, value of $x$

$V=\pi R^{2}H,\pi (R+x{)}^{2}H=\pi R^2(H+x)$;
$R^{2}H+2RxH+x^{2}H=R^{2}H+R^{2}x;2RxH+x^{2}H=R^{2}x$; since $x\ne 0$,
$2RH+xH=R^2;x=\frac{R^2-2RH}{H}$

For $R=8$ and $H=3,x=\frac{16}{3}$, so that (c) is the correct choice.