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# Optimization: area of triangle & square (Part 1)

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.B (LO)
,
FUN‑4.B.1 (EK)
,
FUN‑4.C (LO)
,
FUN‑4.C.1 (EK)

## Video transcript

let's say that I have a hundred meter long wire so that is my wire right over there and it is 100 meters and I'm going to make a cut someplace on this wire and so let's say I make the cut right over there with the left section of Y I'm going to obviously cut it in two with the left section I am going to construct an equilateral triangle I am going to construct an equilateral triangle and with the right section I am going to construct a square I am going to construct a square and my question for you and for me is where do we make this cut in order to minimize the combined areas of this triangle and this square well let's figure out let's define a variable that we're trying to minimize or that we're trying to optimize with respect to so let's say that the variable X is the number of meters that we decide to cut from the left so if we did that then this this select for the triangle would be X meters and the length for the square would be well if we use X up for the left-hand side we're going to have 100 minus X for the right-hand side and so what would the dimensions of the triangle and the square be well the triangle sides are going to be x over 3 x over 3 and x over 3 is an equilateral triangle and the square is going to be 100 100 minus x over 4 by 100 minus x over 4 now it's easy to figure out an expression for the area of the square in terms of X but let's think about what the area of an equilateral triangle might be as a function of the length of its sides so let me do a little bit of an aside right over here so let's say we have an equilateral triangle we have an equilateral triangle just like that and its sides are length s s and s now we know that the area we know that the area of a triangle is 1/2 times the base times the base times the height so in this case the height we could consider to be an altitude if we were to drop an altitude just like this this length right over here this is the height and this would be perpendicular just like that so our area our area is going to be equal to 1/2 times our base is s 1/2 times s times whatever our height is times our height now how can we express H as a function of s well to do that we just have to remind ourselves that what we've drawn over here what we've drawn over here is a right triangle it's 1/2 it's the left half of this equilateral triangle and we know what this bottom side of this right triangle is this altitude splits this side exactly into two so this right over here has length s over 2 so to figure out what H is we can just use the Pythagorean theorem we would have H squared plus s over 2 squared plus s over 2 squared is going to be equal to the hypotenuse squared is going to be equal to s squared so you would get H squared plus s squared over 4 is equal to s squared subtract s squared over 4 from both sides and you get h squared is equal to s squared minus s squared over 4 now to do this I could call s squared I could call this for s squared over 4 just to do just to be able to have a common denominator and for s squared minus s squared over 4 is going to be equal to 3 s squared over 4 so we get h squared is equal to 3 s squared over 4 now we can take the principal root of both sides and we get H is equal to the square root of 3 times s over 2 so now we can just substitute back right over here and we get our area our area is equal to 1/2 s times H well H is this business so it's s times this so it's 1/2 times s times square root of 3 s over 2 which is going to be equal to s times s square it's going to be square root of 3 s squared over 2 times 2 over 4 so this is the area of an equilateral triangle as the function of the length of its sides so what's the area of this business going to be so the area of our little equilateral triangle let me write combined area let me do it in a neutral color so let me do that in Y white so the combined area I'll write it a sub C is going to be equal to the area of my triangle the area of my triangle a sub t plus the area plus the area of my square well the area of my triangle we know what it's going to be it's going to be square root of 3 times the length of a side squared divided by 4 so it's going to be square root of 3 let me do that in that same yellow color it's going to be make sure I switch colors it's going to be square root of 3 over 4 times the side squared times x over 3 x over 3 x over 3 squared all I did is the length of a side is x over 3 we already know what the area is it's square root of 3 over 4 times the length of a side squared and then the length of and then the area of this square right over here the area of the square is just going to be 100 minus x over 4 over 4 squared so our area our combined area maybe I could write like this our combined area as a function of where we make the cut is all of this business right over here and this is what we need to minimize so we need to minimize minimize that right over there and I will do that in the next video