Question

look at the formula for the volume of a rectangular prism. how does the surface area change in each case?a) the length is doubled.b) both the length and width are doubled.call the length, width, and height are doubled.

Answer 1

The Surface area of a rectangula prism is given by

`\begin{gathered} S=2(lw+lh+wh) \\ \text{Where } \\ S=\text{surface area} \\ l=\text{length,} \\ w=\text{width} \\ h=\text{height } \end{gathered}`

Part A

When the length is doubled

`l=2l`

The surface area becomes

`S_1=2(2lw+2lh+wh)`

The surface area will change by

`\begin{gathered} S_1-S=2(2lw+2lh+wh)-2(lw+lh+wh) \\ \text{Then } \\ S_1-S=2(2lw-lw+2lh-lh+wh-wh) \\ \end{gathered}`

Hence

`S_1-S=2(lw+lh)`

Therefore

If the length is double, the surface area will change by 2(lw + lh)

Part B

When both the length and width are doubled

Then

`l=2l,w=2w`

Substitute intonthe formula for the surface area we have

`\begin{gathered} S=2(lw+lh+wh) \\ S_2=2(2l*2w+2lh+2wh) \\ \text{Hence} \\ S_2=2(4lw+2lh+2wh) \end{gathered}`

Then

The surface area will change by

`\begin{gathered} S_2-S=_{}2(4lw+2lh+2wh-lw-lh-wh) \\ \text{Then} \\ S_2-S=2(3lw+lh+wh) \end{gathered}`

Therefore

If both the length and width are doubled, the surface area will change by 2(3lw+lh+wh)

Part C

When all the length, width, and height are doubled

The dimensions becomes

`l=2l,w=2w,h=2h`

Substituting into the formula for the surface area of a rectangular prism we have

`\begin{gathered} S=2(lw+lh+wh) \\ \text{Then } \\ S_3=2(2l*2w+2l*2h+2w*2h) \\ S_3=2(4lw+4lh+4wh) \\ \end{gathered}`

Suntracting from the formula, we have

`\begin{gathered} S_3-S_{}=2(4lw+4lh+4wh)-2(lw+lh+wh) \\ \text{Then } \\ S_3-S=2(4lw+4lh+4wh-lw-lh-wh) \\ \end{gathered}`

hence, we have

`\begin{gathered} S_3-S=2(3lw+3lh+3wh) \\ \text{factor out 3, we have } \\ S_3-S=2(3(lw+lh+wh)) \\ S_3-S=6(lw+lh+wh) \end{gathered}`

Therefore

If all the length, width, and height are doubled, the the surface area will change by

6(lw+lh+wh)