A company sells popcorn in cylindrical canisters. Marketing indicates that wider
canisters will increase sales. If the diameter of the canister is increased
by 27% while keeping the volume of the canister the same, by what
percent must the height be decreased? Express your answer to
the nearest whole number.
Let the original diameter of the canister be $d$, and let the original height of the canister be $h$. Then the original volume of the canister is $\pi r^2 h = \pi\left(\frac{d}{2}\right)^2 h$.
If the diameter of the canister is increased by 27%, then the new diameter is $1.27d$. Since the volume of the canister remains the same, we must have \[
\pi\left(\frac{1.27d}{2}\right)^2 h = \pi\left(\frac{d}{2}\right)^2 h,
\]from which \[
1.27^2 = \frac{d^2}{d^2}
\]and $d^2=1.27^2$. This means $d=1.27$. Thus the new height must be $\frac{d}{1.27}=\frac{1.27}{1.27}=\boxed{100}$ percent of the original height, which is a decrease of $\boxed{0}$ percent.