Practice Test
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A composite solid is a solid that is made of two or more individual solids.
See solution.
We want to draw two different composite solids that have the same volume but different surface areas. A composite solid is a solid that is made from two or more individual solids. Let's take a look at an example pair of rectangular prisms that has have same base, a rectangle with sides a and b.
Notice that the volumes of these rectangular prisms are a b c and a b h, respectively. Now let's combine these two individual solids in two different ways to form two different composite solids.
The volume of each of these composite solids is equal to the sum of volumes of individual solids. This means that both composite solids have the same volume.
| Composite Solid | Volume | Surface Area |
|---|---|---|
| I | a b c + a b h | ? |
| II | a b c+ a b h | ? |
Now let's focus on the first composite solid and find its surface area — the sum of all the surfaces of the shape. Notice that the first composite solid is a rectangular prism.
Let's move to the second composite solid. Notice that here, one face has the dimensions ( c- d) and a.
Finally, we will compare the surface areas of our composite solids. If they simplify to the same expression, the figures have the same surface area. If they do not, then the surface areas of the figures are different. Let's do it!
| Composite Solid | Volume | Surface Area | Simplify |
|---|---|---|---|
| I | a b c + a b h | 2* a b+2* a( c+ d)+2* b( c+ d) | 2ab+2ac+2ad+2bc+2bd |
| II | a b c+ a b h | 4* a b+2* b c+ a c+2* b d+ a d+ a( c- d) | 4ab+2ac+2bc+2bd |
We can see that our composite solids have the same volume but different surface areas, even though they are made from the same individual solids. Notice that this is only an example solution — we can think of many other pairs of composite solids with the same volume but different surface areas.