Answer:
• The scale factor of the volume of the bags is 20.
,
• The scale factor of the dimensions of the two bags is 2.7
,
• The scale factor of the surface areas of the two bags is 7.4
Explanation:
Given:
• The volume of the small bag = 2 pounds.
,
• The volume of the large bag = 40 pounds.
We want to determine the ratio by which the surface area of the small bag increases.
First, given the side lengths of two solids:
• The, ratio of the surface areas, of the two solids is ,the ratio of the square of the side lengths,.
,
• The, ratio of the volume, of the two solids is ,the ratio of the cubes of the side lengths,.
Let the side lengths of the large and small bag be x and y respectively.
[tex]\begin{gathered} \implies\frac{x^3}{y^3}=\frac{40}{2} \\ (\frac{x}{y})^3=20 \end{gathered}[/tex]
The scale factor of the volume of the bags is 20.
Next, find the ratio of the side lengths by taking the cube root of both sides.
[tex]\frac{x}{y}=\sqrt[3]{20}=2.7[/tex]
The scale factor of the dimensions of the two bags is 2.7
Therefore, the ratio of the surface areas will be:
[tex](\frac{x}{y})^2=(\sqrt[3]{20})^2=7.4[/tex]
The scale factor of the surface areas of the two bags is 7.4
