Step 1: Concept/Formula
[tex]\text{Volume of a prism = Length x Width x Height}[/tex]Step 1: Given data
First Prism
Height = 2cm
Width = 4cm
Length = ?
Volume = 24 cubic cm
Step 2:
Find the length of the first prism using the volume formula.
[tex]\begin{gathered} \text{Volume of the first prism = Length x Width x Height} \\ 24\text{ = 2 x 4 x L} \\ 24\text{ = 8L} \\ L\text{ = }\frac{24}{8} \\ L\text{ = 3cm} \end{gathered}[/tex]Step 3: Similarity property
[tex]\begin{gathered} S\text{ince the two figures are similar, the ratio of their corresponding sides } \\ \text{are proportional.} \\ \frac{Length\text{ of 1}}{\text{Length of 2}}\text{ = }\frac{Width\text{ of 1}}{\text{Width of 2}}\text{ = }\frac{Heigth\text{ of 1}}{\text{Heigth of 2}} \end{gathered}[/tex]Step 4: To find the width of the second prism, apply the ratio formula above
[tex]\begin{gathered} \frac{Heigth\text{ of 1}}{\text{Height of 2}}\text{ = }\frac{Width\text{ of 1}}{\text{Width of 2}} \\ \frac{2}{5}\text{ = }\frac{4}{w} \\ \text{Cross multiply} \\ 2w\text{ = 20} \\ \text{Divide through by 2} \\ w\text{ = }\frac{20}{2} \\ w\text{ = 10 cm} \end{gathered}[/tex]The width of the second prism = 10cm
Step 5: Find the length of prism 2
[tex]\begin{gathered} \frac{Height\text{ of 1}}{\text{Height of 2}}\text{ = }\frac{Length\text{ of 1}}{\text{Length of 2}} \\ \frac{2}{5}\text{ = }\frac{3}{L} \\ \text{Cross multiply} \\ 2L\text{ = 15} \\ L\text{ = }\frac{15}{2}\text{ cm} \end{gathered}[/tex]Step 6: Find the surface area of Prism 2
[tex]\begin{gathered} \text{Surface area = 2LW + 2LH + 2WH} \\ =\text{ 2}\times\frac{15}{2}\times10\text{ + 2}\times\frac{15}{4}\times5\text{ + 2}\times10\times5 \\ =\text{ }\frac{2\times15\times10}{2}\text{ + }\frac{2\times15\times5}{2}\text{ + 100} \\ =\text{ 150 + 75 + 100} \\ =325cm^2 \end{gathered}[/tex]Step 7; Volume of the second prims
[tex]\begin{gathered} \text{Volume of the second prism = L x W x H} \\ =\text{ }\frac{15}{2}\text{ x 10 x 5} \\ =\text{ }\frac{750}{2} \\ =375cm^{3^{}} \end{gathered}[/tex]