Video Transcript
Express, in terms of 𝑥, 𝑦, and 𝑧, the sum of the surface areas of the two given figures.
We first need to find expressions for the surface area of each shape. Since they are rectangular prisms, also known as cuboids, each face is a rectangle. So, the area of each face is the product of its sides. A rectangular prism has six faces, and the opposite ones have the same area. So, we can double the area of each visible face to find the sum of the areas of the opposite faces.
Let’s begin by calculating the area of the orange face in the first shape. This is equal to 11𝑥 multiplied by seven 𝑧, which is equal to 77𝑥𝑧. As there are two of these faces, we multiply by two, giving us 154𝑥𝑧. Repeating this for the pink face, we have eight 𝑦 multiplied by seven 𝑧, which is equal to 56𝑦𝑧. Doubling this gives us 112𝑦𝑧. Finally, the area of the green face is equal to 11𝑥 multiplied by eight 𝑦, which equals 88𝑥𝑦. And doubling this gives us 176𝑥𝑦.
We can then repeat this process for the second shape. The orange rectangle has area 91𝑥𝑧, the pink rectangle has area 84𝑦𝑧, and the green rectangle has area 156𝑥𝑦. Doubling each of these gives us 182𝑥𝑧, 168𝑦𝑧, and 312𝑥𝑦. We now have expressions for the surface area of both shapes.
Next, we can find an expression for the total surface area by adding these two expressions. In order to do this, we will collect like terms, recalling that two terms are like terms if they have the same variables raised to the same power. Adding 176𝑥𝑦 and 312𝑥𝑦 gives us 488𝑥𝑦. Likewise, adding 154𝑥𝑧 and 182𝑥𝑧 gives us 336𝑥𝑧. Finally, 112𝑦𝑧 plus 168𝑦𝑧 is equal to 280𝑦𝑧.
So, the total surface area is 488𝑥𝑦 plus 336𝑥𝑧 plus 280𝑦𝑧. We can, therefore, conclude that this is the expression for the sum of the surface areas of the two rectangular prisms in terms of 𝑥, 𝑦, and 𝑧 in its simplest form.
