Grades 7-8 Video Solutions 2022-2022_7-8

Video Transcription

Question 25. A builder has two identical bricks. She places them side by side in three different ways as shown. The surface areas of the three shapes obtained are 72, 96, and 102.  What is the surface area of the original brick?  We have our brick. Let's label the different faces as A, B, and C. One of the properties of a brick is that each face is on there twice.  So on the faces that we do not see, there is another A, another B, and another C.  So we know that we are trying to find 2A plus 2B plus 2C, and that will give us our answer. Now, let's look at one of the shapes. This can be labeled as A, C, C, B, B.  Again, remembering that we are only seeing half of the faces.  So we can multiply it all by 2. So 2A plus 4B plus 4C equals 72. Next, if we look at this layout, we have it look like this. And we get 4A plus 2B plus 4C equals 96.  And with the last layout, we can break down A, B, and C like so. And we get 4A plus 4B plus 2C equals 102.  Now, if we add all of these formulas together, we get 10A plus 10B plus 10C equal to 270. We can divide everything by 10 and get A plus B plus C equal to 27.  But as mentioned earlier, we are looking for 2A plus 2B plus 2C. So we must multiply by 2. And with this, 27 times 2 gives us our answer, which will be D, 54.

Video Summary

A builder determines the surface area of a single brick by arranging two identical bricks in three formations, yielding areas of 72, 96, and 102. By analyzing the arrangements and labeling the brick faces as A, B, and C, equations are formed: 2A + 4B + 4C = 72, 4A + 2B + 4C = 96, and 4A + 4B + 2C = 102. Summing these gives 10A + 10B + 10C = 270, simplifying to A + B + C = 27. Thus, the original brick’s surface area is 2A + 2B + 2C, yielding an answer of 54.

Keywords

brick surface area

mathematical equations

geometry

brick formations

surface area calculation