If the pair of triangular prisms are similar, and the surface area of the smaller one is one hundred and ninety-eight yards squared, find the surface area of the larger one.
First, it is stated that these triangular prisms are similar. Similar solids have the same shape. And their corresponding linear measures, such as these two side lengths nine yards and eighteen yards, they are proportional. And their corresponding faces are similar polygons, just how these are both triangular prisms. If you know two solids are similar, you can use a proportion to find a missing measure. So that means for our question, we can use a proportion to find the missing large surface area.
So what is this proportion that we can use? Well, if two solids are similar, the ratio of their surface areas is proportional to the square of the scale factor between them. The scale factor from the smaller prism to the larger prism is nine to eighteen, which can be written like this: using a colon, using words nine to eighteen, or as a fraction nine to eighteen. Now since we said we’re gonna be using proportions to solve, let’s go ahead and use the fraction.
But before we move on, scale factor should always be reduced, and nine-eighteenths can be reduced to one-half. So the scale factor from the smaller prism to the larger prism is one-half. So as we said before, if two solids are similar, the ratio of their surface areas is equal to the square of the scale factor between them, which would be one-half squared.
So we can solve using proportions because we know the surface area of the smaller prism. We can replace the smaller surface area with one hundred and ninety-eight yards squared. And let’s go ahead and replace the larger surface area with 𝑥 because that is what we will be solving for. In order to square one-half, we need to square one and square two. One squared is one and two squared is four.
This means now we need to find the cross product. So we need to multiply one hundred and ninety-eight yards squared times four and 𝑥 times one. Now we multiply, which gives us seven hundred and ninety-two yards squared is equal to 𝑥. And 𝑥 was the larger surface area. So this is our answer. Seven hundred and ninety-two yards squared is the surface area of the larger triangular prism.