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While processes differ somewhat among the three main manufacturers of cricket balls in international matches, all top-quality cricket balls are handcrafted to exact...

Question

While processes differ somewhat among the three main manufacturers of cricket balls in international matches, all top-quality cricket balls are handcrafted to exacting specifications and standards. Their cork cores are tightly wound with layers of yarn and covered with a leather case with a slightly raised seam that is hand stitched. Once finished, the balls are shaped, stamped, and polished. The labor required is highly skilled and a single craftsman can stitch up to eight cricket balls per day

While processes differ somewhat among the three main manufacturers of cricket balls in international matches, all top-quality cricket balls are handcrafted to exacting specifications and standards. Their cork cores are tightly wound with layers of yarn and covered with a leather case with a slightly raised seam that is hand stitched. Once finished, the balls are shaped, stamped, and polished. The labor required is highly skilled and a single craftsman can stitch up to eight cricket balls per day. The yarn used is high-quality linen. Alternatives to leather from cowhide have been tested but found to be of lesser quality. Given this information, what can you conclude about the production function for cricket balls? What is the cost function? (Hint: See Solved Problem $7.5 . .)$



Answers

Cricket chirping rate In Exercise 15 we modeled tem-
perature as a linear function of the chirping rate of crickets
from limited data. Here we use more extensive data in the
following table to construct a linear model.
(a) Make a scatter plot of the data.
(b) Find and graph the regression line.
(c) Use the linear model in part (b) to estimate the chirping
rate at $100^{\circ} \mathrm{F}$ .

$\begin{array}{|c|c|}\hline \text { Temperature } & {\text { Chirping rate }} \\ \hline\left(^{\circ} \mathrm{F}\right) & {(\text { chirps } / \text { min })} \\ \hline 50 & {20} \\ \hline 55 & {46} \\ {60} & {79} \\ {65} & {91} \\ {70} & {113} \\ \hline\end{array}$
$\begin{array}{|c|c|}\hline \text { Temperature } & {\text { Chirping rate }} \\ \hline\left(^{\circ} \mathrm{F}\right) & {(\text { chirps } / \mathrm{min})} \\ \hline 75 & {140} \\ \hline 80 & {173} \\ {85} & {198} \\ {90} & {211} \\ \hline\end{array}$

I can't read a chapter. One problem 51. And this says baseball rules specify that a regulation hauled away no less five ounces and no mohr than five and 1/4 ounces. So the weight greater than that, but no more than 5 to 5 senses. Okay, so what are the acceptable limits and grants? Okay, so five ounces, two grams. We just need to convert this so it's first good two pounds. His advice in easy. This is one pound has 16 ounces in it. I simply have to cancel out. And now we can get grams two pounds really pounds on tops of what pounds on body, and we want it in grams. So there's 1,000 grams in to put 205 pounds, but also a kilogram. So we do this complication. We come out with 55 you come out with 100 and 42 gramps, so five ounces is good for that. So now we want to find 5 5 to 5 ounces is just multiplied by the same things. 1,000 over t part two of five, and this comes out to be 149 grams. So that means our weight has to be crater. They are equal to 100 5 and less than an equal to 149 grades perfect.

The desired motion is pure tradition about the handle grip. Since the grip is have any lenient motion on access through the grip, Ah can be choosing as on axis fixed in an inertial reference frame. Now the peel rotation condition can be the thing. Us. This is the acceleration off the center off mass disc embedded in US Alpha Matt Dying's the distance off descended off moss minus distance off the clip on Big Grip. Over here is 0.505 Mita from the sin from the end off the back, back to the So that's the distance. That's the grip. Now we applying Newton's second law for Goethe, the translational motion off the center off mus c m on the rotational motion about the handle group. So we have neck force to the equal Do Senate forces F ear, and this is equal to mass times sent it off Mass. According to Newton, slow and for dark talk should be equal toe. I, Elsa and I hear is moment of inner self grip times, angular acceleration and the next talk over here is the force applied times the perpendicular distance. Now we use this relation off off force on dhe user at this place and flying back whilst times acceleration off them that off mass times a day is equal to ay, grip, bang. Satisfied. So now we can plug this value this expression off a C M. Over here. So doing that we get, I saw to the equal Do so Alfa back is basically Alfa, So I'm just writing out for you So I'll for things they see and minus the grip on DDE at the sickle times they on their physical Do I grip Thanks. I thought from here began Sol for D Which concert to be I grip all right whilst times they cme minus decrypt and noticed that the alphas cancel out from both sides. So this is the final expression for data we are going we will be using. But before that we need to calculate mm on DDE the moment of inertia after a dip. So let's do that now said on Let's say this is a question one that we will be using here. Sonal loving was calculated the moment of inertia off the back about an access through the drip, the moss off the back. It is in here on the location off the center off Mars, which is D C M. No, I grip and Gilligan as this is basically mass Times square off the radius. And here isn't the mosque, if some changing. So we take a dining element off Mass on Well, to play it with its radius squared off the radius. And therefore, the total moment off initial should be integration over this. Now the M Here, let me just write it in green. The in here is Lambda Times The X The lambda is the senior mass density off. Huh? Lamb buys the linear mass density on out here is X minus the grip. So we take the square that on blood this expression off again on we're integrating form from cedar toe the length off B X, which is Siegel point 84 meters. Now let's substitute the values that we have. So we have a big group to the equal. Go Cedar Point Siegel! Fife, I'm not writing that. I'm not including the units. Now, after I'm done with all the calculation, I'll write the unit for the center off months. So this is that on Lambda is 0.61 this 2.3 times x skirt and then we have the X. Now we're going to expand this whole expression. So we have so far we expanded on Write it in, Oh, descending order off the orders. So, force, we have extra depart for the highest order. And we have Lena function of X and the constant and we multiply this and we integrate this over the ex. Now that we know what what the limits off X began simply do the integral and put these limits on the day Christian On doing that, they get I grip to the equal toe cedar point 33 6865 plays in with their split. So let me just give you a generalized formula for the integration that we'll be applying over here. So if you have to integrate, that's six to the pot and over the edge six and the limb itself from a to B, then the integration off. This is next to the part and plus one over and bless one. Then you apply the limits A to B and therefore this is equal dough one over and bless one. They will depart in plus one minus. I need the departed Bliss one. So we're substituting beyond a very will be have X now over here the initial m a crazy or the lower limit. Ezio, this means that a is Siegel, This is Eagle. Therefore, we have be to depart and plus lan over and placement on. This is the expression that we are applying now. Then we're doing each indication so here we have extra depart for so four so n here is equal to four. So we should have be to depart for over beauty by five over five and be here is Cedar 0.84 on. After doing that after the integration, you multi played with the constant or the component coefficient off that power coefficient corresponding to that power on DDE. We use the same formula for each of these storms. Ah, just at the value off in keeps on changing. So here in his three year. And it's true. And here in this one and here in this city on remember that this formula is only applicable men. And is it vintage so next. So now that it's clear for the integral part, we move on to find the moss. So months is basically integration off all those small element off masters on Dhe. As I drove over here that this D M is equal to land the times, the small element off lend taken with this t x on again, the limits are going to be saying, which is your 0.84 zero does it a 00.84 meters. So I'm not going to do the full integration over here because, wow, it's quite similar to this one. Re plug this value off line now over here and integrate a question so and then applied the limits. So doing that the u get and to be equal toe one point 1644 gauges last. We have refined the position off the center off Mars, which is ex cm or basically BC, and that we have years. So this is equal. Do the boat must, which is signs some mission. And here we have integration over the multiplication off all ex ambience, all beings, all the small element off my cells times the distance again. We do the same. We have Eckstein's lamb dug lines, the X notice that Lambda is a constant so you can't take it out off the integral. So you have to write fishin off. Landau used the expression off lambda that we have. Which is this one? So little interest rate. This is Lambda, and you have to use it over here s mint, and then you'll have a polynomial on. Limits are going, Toby. See you expand the polynomial on Dhe used this integration drawing and solve it. You get the center off, Moss miscalculated. Toby Cedar point 53757 on. Remember to use this value for Mars over here. Now we have all the values that we need. The Solvay question one this box to question here. So we simply put them with those values in the data question. So we have the equal door administrated again. I agree. All life loss times and not in Mr Total. Maas Times position off the center of mass minus big rip. So the gravy? No. So this is basically zero point 05 meter, as I told over here on this stuff, the values which just calculator some on the line in them. So these three have the values on DDE. Simply plug them into this a question to find the on DDE comes out to be a zero point five line. Jeannie are nearly equal to zero point 59 team Doc. So this means the distance from the end off the bag to the so called sweet spot. Then the equal do be less day grip it 0.5 me, Dutch. And if you use this value over here, you've got this value, Toby. 0.643 meter. So this is the distance from the end off the back to the street sport.

So the surface area of the ball is It is for hi r squared. And if you calculate that we see that are is 0.0 toe meter. Put a square there so we get five or three times. 10 to the power. Negative. Three meters squared now using a Christian 18 0.37 Ah, with t I being equal to 35 plus 2 73 Kelvin. Uh, it is 308 Kelvin on da tf being equal toe 47 plus 2 73 Kelvin, which is so you, 20 Kelvin. The power required to maintain maintain the temperature with the PR, which is equal to Alfa at Salon. Um mmm. Times do you do the bar f to do a four d after the bar for my nasty eye to the bar for so no. Ah signifies constant, which is 5.67 times. Tend to the bar minus eight. What for? Meter squared. Kate and bar four. Excellent for in our case is 0.8 and ah, TF is running on a space here. So TF is 3 20 bucks 3 20 kelvin to the par four minus t 08 Kelvin to the par four so that someone duplication with the upper term. So combining this, we see that, um ah, the power is zero point t four. What now does the heat, um, each be must produce during 20 minute interval Will be Q over end because ah, and is that whole number of peace and ah, pr times t is the total amount of heat that's produced so we can divide that by total number of bees. And that gives us zero pointy For what, times? 20 minutes. We convert that two seconds, So that's gonna be 60 seconds minus divided by one minute, divided by 500. It is a number of peace, So that gives a 0.81 Jules. All right, hold that help.

We have been given two points. We know that given two points, we can find the slip given the equation. Why two mice? Why? One over x demise X one. So 80 minus 70 over 1 68 minus 1 20 Gives us a slope of five over 24. Now he confined the equation. The land given the equation. Why minus y one equals M times X minus X one. This is point slope form equation. You know what this simplifies to five AKs minus 24. Why equals negative? 10 80. This is the answer to part now. Part beak. Given we're looking at chips X equals 1 50 we would plug in for 1 50 Putting X equals 1 50 see Question. We just drived equals negative. 1080 and you end up with why equals 76.25 degrees Fahrenheit. So this is the estimated temperature. This is the answer to part B. S room temperature be 76 point to five degrees Fahrenheit.


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