We want to find the surface area of the smaller solid. Let's begin by finding the scale factor from the larger solid to the smaller solid. The ratio of the side lengths of their bases will give us the scale factor.
Scale Factor
14/12=7/6 ⇔ 7:6
The scale factor of the solids is 7:6. Notice that we have been given just enough information to find the surface area of the larger solid. Therefore, let's first find the surface area of the larger solid by using its open form.
As we can see, the solid consists of four triangles, four rectangles, and one square. Let's find the area of the triangles, area of the rectangles, and the area of the square.
Triangles: 4( 14* 152)&= 420 cm^2
Rectangles: 4(12* 14)&= 672 cm^2
Square: 14*14&= 196 cm^2
Now, we will add them together to find the surface area of the larger solid.
420+672+196=1288 cm^2
Next, we will recall Theorem 12.1 to relate the scale factor to the ratio of the surface areas.
Theorem 12.1
If two similar solids have a scale factor of a:b, then the ratio of the surface areas is a^2:b^2, and the ratio of the volumes is a^3:b^3.
By this theorem we can find the area scale factor of the solids.
c|c
Scale Factor & Ratio of the Surface Areas
7:6 &49:36
Now we can write a proportion to find the surface area of the smaller solid, S.
1288/S=49/36
Finally, let's solve this proportion for S.
