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12.6 Notes

Added May 06, 2015 •

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okay we're going to start chapter 12 but I went ahead and jump to 12.6 about surface area and volume of a sphere because it's only one shape in case I'm not here so first of all surface area surface area is just the amount of space that is taken up on the outside of the where does not contain what's inside but the total surface area of the outside the area that's covering it volume is how much space it has on it inside of the sphere so it's the total amount of stuff that can be inside of this object that would fill it up so first of all is fear is a set of points in space that are equidistant from a given point called the center so so just like a circle is a set of points that are the same distance from the center so is a Seer but it's in a 3D object and not just 2D alright so this is what are sphere looks like right here a globe of all whatever you want to call it now it has several parts and just like a circle dozen most former name the exact same thing radius diameter chord in that kind of stuff so if you look at the first thing is you got to know what the radius is the radius is the segment from the center to a point on the sphere just like they're going to be an infinite number in a circle there can be an infinite Uranus fear but it's still going to be from the center of the object to any point on the circle so there's a radius right there I can draw another one where is here and of course we're looking at it too deep but it can be going towards you away from you you can be going at a diagonal but from the center of that ball to any point on the surface now chord is just like a chord in a circle but it's three dimensional instead of just two dimensional its segment whose endpoints are on the sphere again so just like a regular circle into deform so cord be from here to hear that would be your court and of course it can go front back side to side and any direction as long as it ends on the circle  and then there's diameter and of course that's a chord that contains the center of the circle please are all the exact same definitions it's just 3D instead of 2D so again right here is our diameter and it has to go through this it can go from top to bottom side to side in any direction  now here's a little information that you may not know where maybe a little new and every spear has what they call it a great circle now the great circle is just the intersection of a plane and a sphere that contains the center of the circle so basically it's taking the ball and it's cutting in and 1/2 here's a solid model that you can see so it's like half the globe so think about it like it's the equator Globe is what that is so it's right in the center in it cut it in half so it would look like this so this little circle right here that is called a great circle it's the equator type thing on a glow but now it see this is just the plane that cuts through at that point except this right here would be your great Circle K now there's two things to know about a great circle when you ask for the circumference of sphere  it is going to be the circumference of the great circle is the circumference of the sphere so you don't take appear you have to take it right in the middle of that is the circumference of a sphere it has to go through that great circle which is in the very center so that's considered the circumference not up here and also a great circle separates into two congruent is called what do you call half of a sphere a hemisphere and you have heard that word many times before 2  okay now we know all the parts of a sphere there's only two formulas you need you need one for surface area one for volume so this is your formula for surface sing this is your formula for volume now one thing you do need to remember surface area is still just area it is going to be measured in squared units feet squared to meters squared meters squared and volume is a three dimensional measurement so it is going to be measured in huge units and we'll go over this little more in detail next week Terraria all your answers will be in square units volume all your answers will be in huge units so there's just a couple examples okay so here's a few examples of all of this and it'll be like your homework number one name the center of this sphere will the center right where is T so the center here is T and then it says name a cord will cord goes from  1 + .2 + I'm Circle so it could be QR PS is Accord here or you can name an arc you so Arc you is one exam PS is another one I'm a go ahead and ride them all  number 3 name a segment that is a radius in this sphere radius has to start at the center go to app pronounce that eqt STP those are all radii so cutie AT&T s now you notice they all have tea in them because that's where it has to start at that says name a segment that is a diameter and there's only one show PSR SP  and you notice those are all line segments not lines and the number 5 Conference of the great circle that contains pns write your answer in terms of Pi now it says in terms of Pi that means you just leave the pie you don't actually multiply but the great circle has to go  in the circle that cuts it in half so you're great circle that goes through PS is going to be this one right here now circumference is 2 pi r or Pi D they give us the radius right here so we'll use two times pi what times are circumference is 2 pi are so circumference is 2 * pi * 7 now if they want us to leave it in terms of hi we don't multiply that just multiply the two in the 7 so that's 14 pie now if you wear them all to fly that out you would 43.98 meters so that would be and you do need to put your units that circumference so it's single units next number six it says find surface area and area okay so all you do is use your two formulas if you'll look here you'll notice that my diameter of the spheres 15 so think about it my diameter is 15 yards what is my raise going to be able to be half of that so it's going to be 7.5 and that's what we're going to use in both of our formulas alright so surface area first list of surface area I do essay but they just do **** and it doesn't matter which one you do is 4 times pi times the radius squared so it's four times pi your radius is 7.5 squared type that your calculator exactly like it looks 4 * pi * 7.5 squared and you are going to get 706.86 area so it's yards squared in that is your surface area K volume you're just going to use this formula 4/3 pi R cubed so you'll do 4/3 * pi your radius is 7.5 Ice Cube no type it in your calculator exactly like that however there's one thing when you type this in your calculator you need to put the 4/3 currency then put times pi  times 7.5 in to get a queued on your calculators you're going to have to use the house top button so if you'll do the little house top three that will keep it type in your calculator exactly like that can you were going to get to 35.62 yards volume is cubed soap your volume children 35 cubic yards in your surf series 700 in 6 yards squared  okay the next set in exercise after 15 use the diagram the center of the sphere is C and the circumference is 17 Pi feet so that circumference is this distance around here is 17 Pi feet okay so it says what is half of the sphere called will what's half of a sphere it's a hemisphere so you'll just right hemisphere  can the number 13 says find the radius well you know the circumference correct if you know the circumference you can find the radius and I'm going to write it over here circumference is 2 pi r and we know the conference is 17 Pi so that equals 2 * pi * R and you just want to solve for that are and get by itself well you need to get rid of the two in the pie here and the two in the pie those two cancels the pies cancel so our equals now look here your pies go ahead and cancel so just do 17/2 and you're going to get 8.5 okay look at you also know circumference is pi D so if you would get this you know that's pi x a Diener the diameter 17 right so the this is going to be half of that half of 17 is 8.5 so right here are radius is 8.5 and the says feet pure Sophie the diameter is going to be double that which is 17 feet and number 15 says find the surface half of the sphere so fine the entire surface area and just divided by 2 so if we do that surface area is 4 pi r squared so do 4 x Pi a radius right here is 8.5  squared type that in your calculator exactly like it looks you get 907.29 that's your surface area now you only want half of it so divide by 2 to get the hemisphere and you will have 450 3.96 surface area area so its feet squared  okay in the last two problems they give you a surface area and then give you a volume and they say find the radius well if you look in your are they both have a radius in it so this one is surface area will the formula for surface area is 4 tons times r squared  s equals that amount so put that in 4S that is your surface area so 24 pi equals 4 pi r squared not remember you want to get the radius so you got to get rid of the hi so what you going to do is divided by 4 Pi here 4 Pi now that cancels in that cancels in your left with ours turn in on this side go ahead and cancel that your pies and it's just three 24/4 that is 81 now you have a radius where to take the square root of each side and your radius is 9 we're in centimeters centimeters so your radius is 9 cm just plug in what your surface area into the surface area formula Now volume same exact thing your volume formula is volume equals 4/3 pi R cubed well your volume is 36 Pi so 36 hi equals 4/3 pi r squared now there's several ways that you can go about doing this I would get you one I would get rid of this for over 3 first and the way that you get rid of a fraction as you multiply by the reciprocal of the side so if I take this side and I X me up at 4 and I take this side and I X 3 over for that 3 cancels with that 3 in that for cancels with that for and therefore you are left with hi guys I am so sorry I messed this up record that should be at 3 that should be A3 it's cubed not squared so that's cute okay and then do 3/4 x 36 and the pie will stay right there so 3 force x 36 you can do 3 * 36 / 4 or you can do 3/4 in a parentheses x 36 either way you do it you're going to get 27 okay you'll have to get this radius Cube by itself so we need to get rid of this pie so divide both sides by pi that cancels that cancels so are you have are Cube equals 27th okay when you have over here this r squared you take the square root will there is such thing as a cubed root and I know we haven't done a lot of that but if you go on your calculator and you will hit the math button if you look number for is a cubed root symbol so what you do is you take the cubed root which has a 3 right of each side in those cubes cancel each other in you just get are now the cubed root of 27 and its what play by itself 3 times is 27 and if you take the cube root it's three three times three is nine nine times three is 27 so 3 cubed 27 so your radius here is going to be our but I own your calculators if you will just find the cubed root function like I said turn our TI-84 plus is you hit the math button and it's the number four on that is your cue great you'll hit that button and then type in 27 in hit-and-run okay that wraps up this lesson

Uploaded by Pam Chastain.

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12.6 Notes

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