Question

A food company sells its corn flakes in boxes of two different sizes: the regular box and the family value box. For the family value box, the length of the box has been increased by 25%, the height has been increased by 15%, and the width remains the same. By what percentage does the volume of the box increase from the regular box to the value box? Round your answer to the nearest percent.

Answer 1

The volume of recangular prism can be calculated as:

`V=w\cdot l\cdot h`

V: volume

w: width

l: length

h: height

The box heigth has been increased by 15%, this means that to the original heigth 0.15% of it has been added:

Let "x" represent the original height of the box

`\begin{gathered} h=x+0.15x \\ h=1.15x \end{gathered}`

The box length has been increased by 25%, this means that to the original length, 0.25 more has been added

Let "y" represent the length of the box:

`\begin{gathered} l=y+0.25y \\ l=1.25y \end{gathered}`

Let "z" represent the original width of the box.

The original volume of the box can be calculated as:

`V_{\text{old}}=xyz`

And the new volume of the box can be calculated as

`V_{\text{new}}=1.15x\cdot1.25y\cdot z`

To calculate the percentage increase you have to subtract the old volume from the new one and divide it by the old volume.

`\begin{gathered} \text{Increase}=\frac{V_{\text{new}}-V_{\text{old}}}{V_{\text{old}}}_{} \\ \text{Increase}=((1.15x\cdot1.25y\cdot z)-(xyz))/(xyz) \\ \text{Increase=}(2.4xyz-xyx)/(xyz) \\ \text{Increase}=(1.4xyz)/(zyx) \\ \text{Increase}=1.4 \end{gathered}`

The 1 represents the original volume, so that the box volume was increased 0.4 of its original volume.

Multiply it by 100 and the percentaje is 40%