in this video we're going to look at the unit 2 lesson one practice problems for algebra 2. number one we have a rectangular schoolyard is to be fenced in using the wall of the school for one side and then 150 meters of fencing for the other three sides so I like to just start by kind of drawing a picture of this so if we think of this as the school wall um and then we're gonna fence in a rectangular schoolyard okay so something like this we're gonna have the fence drawn around here using the school as one of the sides so then it says the area of um a of X is the square meters of the schoolyard okay so this area here is a function of the length x in meters of the sides um and so if we just call each side here x so this side um X and then if we called this side x because we don't know these lengths but we do know that these two sides are the same length so we'll call them the same variable X then how long would this third side be so remember we have a 150 total meters of fencing and 2x of that fencing is used up so we would take the total 150 and we would subtract off the 2X and that's going to be this piece of fencing that's left so then when we go ahead and write the expression for the area you'll remember um that the area formula for a rectangle is going to be the length times the width so we're going to do in this case the length or width which is X okay and then times the length or width which is 150 minus 2x so we'll just multiply these two pieces together so this would be an expression for the area then it wants to know what would the area be when X is 40. so this means that we're going to plug 40 into this function for X um so everywhere I see an X I'm going to put in 40 okay so we're going to put in 40 right here for x and then we have 150 whoops 150 minus 2 times x okay or 40 in this case so we're just plugging 40 in wherever we see X so then we'll go ahead and simplify this so we're going to get um 40 times and then this is going to be 80. and 150 minus 80 and that parentheses is 70. so 40 times 70 is 2 800 and then this is in meters so the area is going to be meters squared so what's a reasonable um domain in this context so when we're taking a look here we're not going to want and let me change this color we're not going to want this to go negative because we can't have a side length where this would go negative so this we have to make sure stays above zero and so 150 minus 2 times what would give us zero so you can certainly go ahead and solve this um and set it equal to zero you can also think you're just going to divide 150 by 2. okay so I'm just going to add 2x to both sides here um so we get 150 equals 2x so divide by 2 and so if we have x equals 75 that would make this side zero and so we don't want to go any more than that so we can't go less than zero because we have to have something here and then we want our X to be less than that 75. so we want it to be greater than zero and less than 75 for the length of um X on this fence number two Noah finds an expression for V of X so that's going to be volume um that gives the volume of an open top box in cubic inches in terms of the length x of the cutout of squares used to make it in this graph Noah um this is the graph Noah gets if he allows the values between negative one and five so what would be a more appropriate domain for NOAA to use instead so remember that this um vertical axis here is the volume okay so remember we're calculating volume so the height of this function represents the volume so we wouldn't want to go um below this horizontal axis because this horizontal axis would be where our volume equals zero so anything below that is a negative volume so we can see that this function or the volume is below zero here okay at x equals zero and then it dips below here so once it goes below zero then that's going to be bad so this is about x equals 2.5 so nothing after that so we want to just be in between these X values so we want to be um zero is less than or equal to X is less than or equal to about 2.5 so that's going to be um The Domain we want between 0 and 2.5 so what does it look like the maximum volume is going to be so this is going to be the highest volume so we see here and we look over and that shows that our max volume is about 15. number three my wants to make an open top box by cutting out corners of a square piece of cardboard and folding up the sides the cardboard is 10 by 10 centimeters and then we're going to do the volume in cubic centimeters to open the box as a function of a single length x so we're going to want to write an expression for this so we're going to start with a square because we know that my is cutting out of a square this square is 10 by 10. and then mine's gonna cut out um squares from the corners okay so from each of the corners she's going to cut off this amount now we don't know exactly how much she's gonna cut off so we're gonna account for that by calling the amount that she cuts out X so let me just get this into each corner here so then this will give us kind of an open side here but this is going to be x x x okay so we're cutting out an X by X Box from each side okay so this is getting cut out so write a ball write an expression from the volume so then this is going to fold up so this is going to be our height so remember when we do volume we do length times width times height okay so we're going to want to multiply um kind of this base here that we're going to get when we fold it up so if you can imagine um these being cut out here so these are going to fold up so these don't exist these got cut out okay and then we're gonna fold you know along here to create this box so the the length here okay is going to be we started with 10. and now we've subtracted in an X here and an X here so this length is going to be 10 minus 2X same with this side okay 10 minus 2X so we're going to do the length times the width and then times the height of the box so if you fold it up this would be the height so then times x so what is the volume of the box when x equals three so now we're just going to plug 3 in here so we're going to do the volume of three so we'll do 10 minus 2 times 3 which is 6. so then 10 minus 2 times 3 which is 6. and then times x which we know is 3 in this case so then 10 minus 6 is 4 10 minus 6 is 4 so we're going to do 4 times 4 times 3 so we're going to get 16 times 3 which is 48 centimeters cubed for the volume number four the area of a pond covered by algae is one-fourth of a square meter on day one and it doubles each day complete the table so the algae is 1 4 square meters on day one then we're going to multiply by two so remember when we multiply by 2 so 1 4 times 2 okay is like two over one so that's going to double the top and then four times one on the bottom this simplifies to one half one half times two is one whole one times two is two two times two is four four times two is eight number five we have a table of values for the sequence P Define p recursively using function notation so we want to see what's happening okay as we move and we can see that that's a pretty significant drop so it's not falling at the same difference we're not subtracting the same thing okay but we are dividing by ten so we're losing a zero each time so that's dividing by ten or multiplying by one tenth so to define recursively we need to say the first term which is five thousand then we need to Define for the nth term okay which is going to be 1 10. of the N minus one term or of the previous term so this thing right here means previous term and then we would say for n that's greater than or equal to two since we're defining the first term then we'll start this sequence by plugging in for the second term and higher all right and number six it says the table shows two sloth populations growing over time okay then ask some questions about it so the first one is to describe the pattern so see if um population one looks to be decreasing okay so it's decreasing so it's either minusing the same number or it's um decreasing by a common factor and if we look here okay it kind of does follow a pattern until this three um so we know we're not subtracting if we're going zero five zero three zero so we can try for a common ratio so we're going to divide the new population by the original and when you do Point uh when you do 76.5 divided by 90 you'll get 0.85 so 90 times 0.85 is 76.5 76.5 times 0.85 is 65. 65 times 0.85 is 55.3 so it appears to be following this pattern okay and then 53 times 40 uh sorry 55.3 times 0.85 is 47. um so figure out this one in a second but part B asks us to continue this pattern so let's just continue this one while we're working on it so keep multiplying by 0.85 you'll get 40. and then 34. then 28.9 and then 24.6 so then in this second sloth population okay we see we go 39 to 37 then to 35 33 31 so hopefully we see that pattern of subtracting 2 so if we continue that on we're going to get 29 27 25 and then 23. so then part C wants us um on an axis to graph both of these populations okay so let me just graph get some axes here okay and then it wants us to graph both populations on here now um the first population started at 90 so that's the highest we need to get so I'm just going to put this at 100 and halfway down we'll put at 50. and then we need to go um at least up to eight because that's what our table is on this x-axis so that's three four five and then six seven eight and then we're just gonna plot those points so in this first table um it had us at zero we were at 90. one we were at 76.5 so about halfway between 50 and 100. two after two years we were at 65. after three years we were at 55.3 after um four years we were at 47 so we dropped below that 50. five years we were at 40. six years we were at um 34 so over halfway still between 0 and 50. um six years we were at 28.9 seven years we were at 24 okay so you can see that this is curving but slowly okay curving down um then we're gonna graph the other population that one started at 39 and went down by two each year so the first year we were at 39 I'm sorry the zero year we are at 39 then we were at 37 then we're at 35 then we were at um 33 then we were at 31 29 and I'm a little bit low on this let me get um let me get the twin we're about where 25 would be here okay so this is where 25 would be halfway between so let me plot these again so at zero we'd be at 39. at one we'd be at 37 at 2 we'd be at 35 at 3 we'd be at 33. okay at four we'd be at 31 at five we'd be at 29. at six we'd be at 27. at seven we would be at 25 okay then at eight we'd be at 23. so now Part D asks will the two populations ever equal and if so when so we can see that they're about to touch okay so it's going to be pretty close here so let's just do one more um year so let's look at year nine this one would go down to 21 and if we multiply this by 0.85 this will be at 20.9 which is nearly 21. okay so this is the populations are going to be equal at approximately or about nine years