Polynomials word problems
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Find the volume of a tank whose base has an area of 3x squared plus 30x plus 5 square feet and whose height is 8x minus 5. So let me draw a potential tank here. Maybe it's a cylinder, so let me draw a cylindrical tank here, just like that. It's supposed to be a cylinder, and maybe if it was transparent, you would see this back side over there. And these two are the same. These are supposed to be straight lines. And the volume of a-- let me just label it first. So the area of the base, which is the same as the area of the top here or of the base here, the area of that is 3x squared plus 30x plus 5. And they tell us the height is 8x minus 5. The height of this tank is 8x minus 5. And if you want to find the volume of a three-dimensional object like this, you just multiply the area of the base times the height. So the volume is going to be the area of the base, which is 3x squared plus 30x plus 5, times the height, times 8x minus 5. Now, to multiply something like this out, it might seem really complicated, but we once again, we just have to do the distributive property. If you viewed this big thing in pink here as just a number, if this was like the number 7, you'd just say, well, this is going to be 7 times 8x minus 7 times 5 or minus 5 times 7. You would just distribute it out. You would just multiply this entire thing times each of the terms. That's what the distributive property tells us when you first learned it. So let's do that. So it's going to be this entire thing times 8x, or we can view it as 8x times this entire thing. So 8x times this entire thing: 3x squared plus 30x plus 5, minus 5 times the entire thing again, or the entire thing times minus 5. So once again you get 3x squared plus 30x plus 5. And now we just multiply these out. We just distribute out the 8x over this whole thing and we distribute out the negative 5 over that whole thing. So then we get 8x times 3x squared is 24x to the third power. 8x times 30x is what? That's 240x squared, so plus 240x squared. 8x times 5 is plus 40x. And then we multiply this negative 5 out. Negative 5 times 3x squared is negative 15x squared. Negative 5 times 30x is negative 150x. And then negative 5 times 5 is negative 25. And then we just have to simplify it from here. So we only have one third-degree term, one thing that has an x to the third in it. We have this term right here, so we'll write that as 24x to the third. And then what are our x squared terms? We have 240x squared minus 15x squared. So what's two 240 minus 15? It's 225x squared. So plus 225x squared. That's adding that term to that term right over there. And then we have 40x minus 150x. That would be negative 110x. And then finally we have just this negative 25 out here. That's the only constant term. And we're done! We found the volume of the tank. It's given by this polynomial expression right here. So this right here is the volume of the tank. It's equal to 24x to the third plus 225x squared minus 110x minus 25.