Exercise 23 Page 647

Now we want to find the ratio of the surface areas and volumes of the given cylinders. Let's start with the ratio of the surface areas. To do so, we will recall the relationship between the surface areas of similar solids.

Surface Area of Similar Solids

If Solid $X$ is similar to Solid $Y$ by a scale factor, then the surface area of $X$ is equal to the surface area of $Y$ times the square of the scale factor.

Therefore, to get the $\text{surface}$ $\text{area}$ $\text{of}$ $\text{Cylinder}$ $\text{A},$ we need to multiply the $\text{surface}$ $\text{area}$ $\text{of}$ $\text{Cylinder}$ $\text{B}$ by the $\text{scale}$ $\text{factor}$ raised to the $\text{second}$ $\text{power}.$ In Part A, we found that the ratio of the radii of the cylinders is $3:1.$ This means that the $\text{scale}$ $\text{factor}$ is $3.$ We can rewrite this equation to get the ratio of the $\text{surface}$ $\text{area}$ $\text{of}$ $\text{Cylinder}$ $\text{A}$ to the $\text{surface}$ $\text{area}$ $\text{of}$ $\text{Cylinder}$ $\text{B}.$ The ratio of the surface areas of the cylinders is $9:1.$ Next, we will find the ratio of their volumes. To do so, we will recall the relationship between the volumes of similar solids.

Volume of Similar Solids

If Solid $X$ is similar to Solid $Y$ by a scale factor, then the volume of $X$ is equal to the volume of $Y$ times the cube of the scale factor.

Therefore, to get the $\text{volume}$ $\text{of}$ $\text{Cylinder}$ $\text{A},$ we need to multiply the $\text{volume}$ $\text{of}$ $\text{Cylinder}$ $\text{B}$ by the $\text{scale}$ $\text{factor}$ raised to the $\text{third}$ $\text{power}.$ We can rewrite this equation to get the ratio of the $\text{volume}$ $\text{of}$ $\text{Cylinder}$ $\text{A}$ to the $\text{volume}$ $\text{of}$$\text{Cylinder}$ $\text{B}.$ The ratio of the volumes of the cylinders is $27:1.$

In Part B, we found that the ratio of the surface areas of the cylinders is $9:1.$ Since we know the surface area of Cylinder A, we can calculate the surface area of Cylinder B using this ratio. Let's do it! We got that the surface area of Cylinder $B$ is $602.88{\text{ cm}}^2.$

In Part B, we found that the ratio of the volumes of the cylinders is $27:1.$ Since we know the volume of Cylinder B, we can calculate the volume of Cylinder A using this ratio. Let's do it! We got that the volume of Cylinder $A$ is $30520.8{\text{ cm}}^3.$