Polynomial multiplication word problem

find the volume of a tank whose base has an area of 3x squared plus 30x plus five square feet and whose height is 8x minus five so let me draw a potential tank here maybe it's a cylinder so let me draw a cylindrical tank here just like that just like that supposed to be a cylinder and then maybe if it was transparent you would see this backside 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 the base here the area of that is 3x squared plus 30x plus five and they tell us the height is 8x minus five the height of this of this tank is 8x minus five the height is 8x minus five 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 five times the height times 8x minus five times 8x minus five now to multiply something like this out when you have you know it might seem really complicated but we once again we just have to do the distributive property if you view this big thing in pink here is just a number if this was like I don't know the number seven you would just say well this is going to be seven times 8x minus seven times 5 or minus five times seven you would just distribute it out you just multiply this entire thing times each of the terms so that's what the distributive property tells us when you first learned it so let's do that this is going to be this entire thing times 8x or we could view it as 8x times this entire thing so 8x times this entire thing 3x squared 3x squared plus 30x plus five minus five minus five times the entire thing again or the entire thing times minus five so once again and 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 old thing so then we get 8x times 3x squared is 24x to the third power 8x times 30x is what this 240 x squared so plus 240 x squared 8x times 5 is plus 40 X and then we multiply this negative 5 out negative 5 times 3x squared is negative negative 15 x squared negative 5 times 30 X is negative 150 X and then negative 5 times 5 is negative 5 times 5 is negative 25 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 3rd in it we have this term right here so we'll write that as 24 X to the third and then what are our x squared terms we have 240 x squared minus 15 x squared so what's 240 minus 15 it's 225 x squared so plus 225 x squared that's adding that term to that term right over there and then we have and then we have 40 X minus 150 X that would be negative 110 X negative 110 X and then finally we have just this negative 25 out here we have this negative 25 out there that's the only constant term and we're done we found the volume of the tank it's it's given by this by this polynomial expression right here so this right here is the volume the volume of the tank it's equal to 24 X to the 3rd plus 225 x squared minus 110 X minus 25