Final answer:
To maintain the same volume when the diameter of a cylindrical canister is increased by 27%, the height must be decreased by approximately 43.75%.
Step-by-step explanation:
To determine by what percent the height of a cylindrical canister must be decreased after increasing the diameter by 27% to keep the volume the same, we can use the formula for the volume of a cylinder: V = πr^2h, where V is the volume, r is the radius, and h is the height.
If the diameter of the canister is increased by 27%, the new radius is 1.27 times the original radius. Considering the volume stays constant, the new height must allow the equation 1.27^2 * π * r^2 * new height = π * r^2 * original height to be true. Thus, the new height must be equal to the original height divided by 1.27^2. To find the percent decrease in height, we can use the formula percent change = ((original - new) / original) * 100.
The new height = original height / 1.27^2, and so the percent decrease in height is ((1 - 1/1.27^2) * 100) %. A calculation shows this results in a decrease of approximately 43.75% in the canister's height.