Problem

A food company distributes its tomato soup in two cans of different sizes. For the larger can, both the diameter and the height have been increased by $30 \%$. By what percentage does the volume of the can increase from the smaller can to the larger can? Round your answer to the nearest percent. $\%$

Solution

Step 1 :The volume of a cylinder is given by the formula \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height.

Step 2 :If both the diameter (and hence the radius) and the height of the can are increased by 30%, the new volume \(V'\) of the can will be \(V' = \pi (1.3r)^2 (1.3h) = 1.3^3 \pi r^2 h = 1.3^3 V\).

Step 3 :The percentage increase in volume is then given by \(\frac{V' - V}{V} \times 100\% = (1.3^3 - 1) \times 100\%\).

Step 4 :Calculating the above expression gives a percentage increase of approximately 119.7%.

Step 5 :Rounding to the nearest percent gives a final answer of 120%.

Step 6 :Final Answer: The volume of the can increases by approximately \(\boxed{120\%}\).

From Solvely APP
Source: https://solvelyapp.com/problems/20542/

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