Question

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.

Answer 1

After extending the diameter by 27%, the height needs to be lowered by roughly 39% in order to maintain the same volume.

To solve this problem

Define the variables. Assume that d was the initial diameter.

Let h be the initial height.

Set the volume to V.

Let, set up equations:

• Original volume: V = π(d²/4) * h
• Increased diameter: d' = d * 1.27
• New volume with same original volume: V = π(d'²/4) * h'
• Solve for h': h' = V / (π(d'²/4))

Now, let substitute and solve:

h' = π(d²/4) * h / (π((d*1.27)²/4))

h' = h / (1.27²)

h' ≈ h / 1.64

Let, calculate the percentage decrease:

Decrease = (h - h') / h * 100%

Decrease = (h - h / 1.64) / h * 100%

Decrease ≈ 39%

So, After extending the diameter by 27%, the height needs to be lowered by roughly 39% in order to maintain the same volume.