Ex 2 Surface Area of Composite Objects

determine the surface area of this composite object so if we look at this object we see we have cylinder on the bottom so wrote down cylinder here and we have hemisphere sitting on top so now when I'm referring to the surface area think about if you were to paint this object so what would you what part of the area are or the shape are you painting so we would need the bottom we would need this lateral area area of the cylinder and we would need the latter lateral area of the hemisphere so we don't need the base because the the base of the or the top rather maybe would be better way of saying the cylinder and the base of the hemisphere are touching that part you don't need so the surface area of the cylinder I'm gonna just start with looking at that formula here got that off of our formula sheet right here so wrote it down here for us now the two PI times height is the lateral area in other words that's that area this circular part right here area of the base is 2 PI squared is the top and the bottom but only need part of this I'm only tape painting the bottom so it's two circles that's why it's 2 PI squared so only need one of the circles so when I'm gonna put my formula together for this shape for the surface area I'm just going to use two PI times height so the lateral area plus instead of 2 PI squared just PI squared so this is gonna be the lateral area this is gonna be the bottom ok now let's look at the hemisphere remember that the formula for surface area of hemisphere is 3 PI squared now where did that come from I'm just gonna refresh your memory we had looked at this earlier and surface area is cutting that hemisphere in half so the lateral area is 2 PI squared plus the area of the circle on the top is PI squared well don't need the PI squared anymore because just need the lateral area so I'm not gonna take off the area of the base because the area of the base I'm not painting that part that would be just on top okay so that would mean that this really I'll break this down so we have it here for you this is the lateral area right here and this is the area of the base but in our formula we only need this part so we're gonna say plus two PI squared which is the lateral area of hemisphere all right so we can plug all our numbers in like to try and keep things little bit simpler in terms of grouping things so if have PI squared plus 2 PI squared here like to combine that don't have to can plug the numbers in but see that as 3 PI squared so kind of derive my own formula for my object here ok let's plug in some numbers now so 2 pi radius is given as 2 the height is given as 4 plus 3 PI we're gonna put that radius back in here and square it so let's just simplify this in our head first so 2 times 2 is 4 4 times 4 so we've got 16 pi this allows us to not have to type so much in our calculator and less rent room for rounding errors plus 2 squared you have to do your exponents first is 4 4 times 3 is 12 PI so you have 16 PI plus 12 PI so you can combine those to make this 28 pi this is actually perfectly fine answer in terms of an exact value in other words 28 pi is exactly the right answer of course we want to make little more sense of that and put that into our calculators so if you want your decimal answer 28 PI and your Calcutta 87.9 6 feet squared now if didn't combine this formula and was just trying to get my decimal answer can go ahead and just plug them directly in there just be careful with your rounding that if you are writing these three numbers out individually you write down every little decimal and then punch that back in your target find this lot less work for the calculator piece and more likely to not have rounding error