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Problem 68

A sporting goods manufacturer designs a golf ball…

Problem 67

Jewelry A jeweler resizes a ring so that its inner circumference is 6 centimeters.
(a) What is the radius of the ring?
(b) The inner circumference of the ring varies between 5.5 centimeters and 6.5 centimeters. How does the radius $\quad$ vary?
(c) Use the $\varepsilon-\delta$ definition of limit to describe this situation. Identify $\varepsilon$ and $\delta .$


a) 9.55
b) 0.875
c) $\delta=\frac{\epsilon}{2 \pi}$


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Video Transcript

This is problem number eleven of the Stuart Calculus Eef Addition. Section two point four Machinists required to manufacturer circular metal disc with area one thousand centimeter squared Parting What radios produces such a disc? Don't answer this part. First fifty circular disk. The area is given by pi r squared and since the area is given as a thousand centimeters squared, we just need to solve this equation. One thousand equals pi Times are squared if we take a thousand right by pi And then mom threw that first time was in the bed of my prime. I guess about three hundred eighteen point three This is equal to r squared. So we take the square root and ours approximately Are they approximately seventeen point eight four? And that is Ah, The answer to part a part bean with the machinists is a lot of narrow tolerance of plus or minus five centimeters squared in the area. The disc How close to the ideal radius in part? Jane rest machine is controller radius. So we're going to do is we're gonna take this new information directly and just assumed What the maximum here. Uh, what radius? The maximum error allows. So from the first case, let's say that the maximum in error if it becomes too large is plus five centimeter squared. So our area's a thousand and the five centimeter squared, and we're going to look for, um, this or limiting our So we saw are in the same way as we did in party and this limiting our other way. Yet that is a maximum. Please is about seventeen queen eight eight six for the minimum allowed area. You do the same thing some for the Radius in that case and re carry limiting radius. In that case, off seventeen point seven nine Helen Approximate thing. Therefore, the radius the allow boring yous is somewhere between seventeen, seven, ninety seven and seventeen point of eighty six, right, because in this case, he's already I at the end points, give the minimum and maximum allowable area any release in between us allowed. Notice that the radius and party sits directly between these two numbers and pouring into now is just comment on how close to the ideal radius and partying was. A shyness machine is controller radius so relative to the radius that we found the party. We're going to rewrite this, as are minus So the radius minus the ideal radius seventeen wait for And we see that if we subtract seventeen point eight four two each of these terms, we should can't the maximum seventeen point six six, seventy nine, eighty eighty six minus seventeen point four I know. Or six and then seventeen point seven nine seven minus seventeen point four. Well, yes, it gives up negative zero point. Oh, for three. We see that the the closeness to the ideal radius must be within went on for six and pointed for three. And this seems familiar to us in order to Karen team that we meet there appropriate area and to just allow for one choice of intolerance. Air tolerant on the radius. We're going to chase a smaller have so valley of the two numbers. So we are going to choose point oh four and were any use delta as our air tolerance for the radius point Oh four three. That's kind of point. Our ends of report that is how close to the Yorker radius round. The machinist must control in order to meet the required direction for the circular metal disk and finally in party. We're going to make a comment about all of these parts and how they tie into the limit. Definition two, The first question are in terms of the Absalon to finish delta definition of luminous experts save effects equals l. What is X? Well, we can rewrite or we can rate this problem as the limit is Are the radius approaches the ideal radius, which in our case was seventeen point four No, the area function as a function of our which, if you can recall our area function is pi r squared. And this limit goes towards what we want. This function to become as armor to the ideal radius. And that is one thousand. So what is X from the original definition X is the radius are What is that Alex as the next? A functionary using to determine these are tolerance is the area function Hai r squared? What is a is the look Val, you exit the protein or are the protein are supporting the ideal really is seventeen twenty for pope answer to party. What is hell? Delimit valley? Let me values equal to a thousand. What family of Absalon is given Absalon was thie tolerance that was keeping plus or minus. I have centimeter squared. And for the forthis value, Absalon, we found corresponding bunnies or delta ha to be a corresponding value. Felt it to be about point Oh, for three. And that answers all parts of this problem.

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