Solving optimization problems
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Optimization: area of triangle & square (Part 2)
Where we left off in the last video, we had come up with an expression as a function of x of our combined area based on where we make the cut. And now we just need to figure out where this hits a minimum value. And to do that, we just have to take the derivative of this business, figure out where our derivative is either undefined or 0, and then just make sure that that is a minimum value, and then we'll be all set. So let me rewrite this. So our combined area as a function of x, let me just rewrite this so it's a little bit easier to take the derivative. So this is going to be the square root of 3 times x squared over-- let's see, this is 4 times 9. This is x squared over 9. So this is going to be 4 times 9 is 36. And then over here in blue, this is going to be plus 100 minus x squared over 16. Now let's take the derivative of this. So A prime, the derivative of our combined area as a function of x, is going to be equal to-- well, the derivative of this with respect to x is just going to be square root of 3x over 18. The derivative of this with respect to x, well, it's the derivative of something squared over 16 with respect to that something. So that's going to be that something to the first power times 2/16, which is just over 8. And then times-- we're just doing the chain rule-- times the derivative of the something with respect to x. The derivative of 100 minus x with respect to x is just negative 1, so times negative 1. So we'll multiply negative 1 right over here. And so we can rewrite all of that as-- this is going to be equal to the square root of 3/18 x plus-- let's see, I could write this as positive x/8. So I could write this as 1/8 x, right? Because a negative 1 times a negative x is positive x/8. And then minus 100/8, which is negative 12.5-- minus 12.5. And we want to figure out an x that minimizes this area. So this derivative right over here is defined for any x. So we're not going to get our critical point by figuring out where the derivative is undefined. But we might get a critical point by setting this derivative equal to 0 to figure out what x-values make our derivative 0. When do we have a 0 slope for our original function? And then we just have to verify that this is going to be a minimum point if we can find an x that makes this thing equal to 0. So let's try to solve for x. So if we add 12.5 to both sides, we get 12.5 is equal to-- if you add the x terms, you get square root of 3/18 plus 1/8 x. To solve for x, divide both sides by this business. You get x is equal to 12.5 over square root of 3 over 18 plus 1/8. And we are done. At x equals this, our derivative is equal to 0. I shouldn't say we're done yet. We don't know whether this is a minimum point. In order to figure out whether this is a minimum point we have to figure out whether our function is concave upward or concave downward when x is equal to this business. And to figure that out, let's take the second derivative here. So let me rewrite the second derivative of all of this business. The second derivative, well, this was the same function as this right over here. So let me rewrite it. So A prime, the derivative of our combined area, was equal to the square root of 3/18 x plus 1/8 x minus 12.5. The second derivative is going to be square root of 3/18 plus 1/8. So this thing right over here is greater than 0, which means we're concave upward for all x's. Concave upwards, which means for all x's. We're kind of doing a situation like this. So if we find an x where the slope is 0, it's going to be over an interval where it's concave upwards. This is concave upwards for all x. So we're going to be at a minimum point. The slope is 0 right over here. This right over here will be a minimum point. So once again, this is going to be a minimum point. Now, if we actually had a 100-meter wire, this expression isn't too valuable. We'd want to get a pretty close approximation in terms of where to actually make the cut, an actual decimal number. So let's use a calculator to get that. So we have 12.5 divided by square root of 3 divided by 18 plus 1 divided by 8 gives us-- and now we deserve our drum roll. This is 56.5. So this is approximately equal to 56.5 meters. So you make this cut roughly 56.5-- I'll write roughly-- 56.5 meters from the left-hand side. And you will minimize the combined area of both of these figures.