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Numerade Educator

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Problem 6 Medium Difficulty

You are planning to close off a corner of the first quadrant with a line segment 20 units long running from $(a, 0)$ to $(0, b)$ . Show that the area of the triangle enclosed by the segment is largest when
$a=b_{\text { . }}$

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Video Transcript

From the set of floor question, we have Triangle Bipasha News 20 and then to Unknown Monster Remaining Sides of A and B. Let's find the area which is given by dysfunction for capital A equals ah, half base times site. So half a B. Now this is currently function of two variables on. We would like to simplify it to a function of one variable which we will do by using my family. This is fair. So performance is fair for a triangle is a squared plus B squared equals 400 on. We will solve this for be so B is equal to the square root. Ah, full 100 minus a squad. We've learned that by simply taking the a sweat to the right hand side and then taking the positive square root as we're working in the first quarter int on due to or constraints A is a B or always less than or equal to 20 service value insights. Whoever it is positive no properties formula for B into our function for the area which is capital A. So we get capital E equals half lower case a times 400 have a square root of 400 minus X squared. This is now a function of one variable Andi So it's much simpler it'll work with Onda. We want to find when this takes map absolute maximum value. And to do that we need to look at the end point on and the critical points. So look at the end points in this box here. So the endpoints are given a equal syrup on D A. Equals 20 because working the first quadrant so the minimum value zero on Because the love survive possum uses 20 or other sides cannot be longer than 22. To buy factors is fair. When we take these values and put them into the function for capsule A, we actually get that a zero equals a of 20 equals error. Yes, points important to see because it means that the PM points are angry. Function actually takes a minimum value as area can't be negative. So this isn't what we're looking for As we're looking for the maximum value, find the maximum value we look at the critical points on the critical points are given if you first relative after leg in terms of love. Okay, say when we set that equal to zero now to do the same fracture differentiation. We want to use both the French action by parts on bond. The chain rule so different direction by parts. Yeah, but this point is just off has the square root of 400 minus a squared because the single A vanishes and then this goes to plus off. Okay, Does it remain constant Nazi friendship square root. We bring the power of 1/2 down, but then becomes one over the square root 400 minus a squared on. Then this is where we need to utilize the chamber rule because the square root is off a function of bank. So we need to differentiate this function. So differential off 400 minus a squared is simply minus two way. So we now simplify this So the first fraction stays the same. So it's 1/2. I was a square root of 400 minus a squared Andi. Then the second term becomes minus 1/2 a squad over the square root 400 minus a squad and bring the soul of the one fraction it becomes 200 minus a squared over the square root 400 on its A squad. Now this is equal to zero. Andi. This is actually simple when simpler than it looks, because we can remove for the nominator by multiplying both sides, and it gives us 200 minus. A squad equals zero on by souls. When a equals you square it. It's 200 now to see that this is the absolute maximum we need to put person for formula. For a I'm doing so receive at the Capitol a off 200 that the square root 200 is equal to 100. So this is clearly larger than the value of the area. The points Andi so critical point is the maximum. So we now just need to see what value be takes at this maximum. But we can simply use the pie fibrous fair of solution we have before that B equals the square root 400 minus a squared, but a sweat is equal to 200 and 400 minus 200 is simply 200 so b equals the square of 200 which is equal to a