5

Hardware store carries very popular Ioal blower. However, the store Oyner doos not have Ioom to keop Iarge inventory ol extra Ioaf blowers. Autumn approaching and I...

Question

Hardware store carries very popular Ioal blower. However, the store Oyner doos not have Ioom to keop Iarge inventory ol extra Ioaf blowers. Autumn approaching and Ine ownar neods to decide the minimum and maximum inventory t0 have slock [0 meel consumer needs bolh low- and high-demand umes.How would quantitative analysis improve decision making?The owner can bulld model Ihat detenines Ihe ellect seasonallty on demand: The owne wIIl know exacily how many blowers t0 sell during each period_ The O

hardware store carries very popular Ioal blower. However, the store Oyner doos not have Ioom to keop Iarge inventory ol extra Ioaf blowers. Autumn approaching and Ine ownar neods to decide the minimum and maximum inventory t0 have slock [0 meel consumer needs bolh low- and high-demand umes. How would quantitative analysis improve decision making? The owner can bulld model Ihat detenines Ihe ellect seasonallty on demand: The owne wIIl know exacily how many blowers t0 sell during each period_ The Owne will find the correlation between sales and seasonality: Tho ownc can find the causc-and-effect rclalionship betrcen salcs and scasonality:



Answers

A
store
manager
wants
to

know
the demand
for
a
product
as
a
function
of
the
price.
The daily
sales
for
different
prices
of
the
product
are
shown in
the
table.

(a)
Find
the
least
squares
regression
line
$ y = ax + b $ for
the
data
by
solving
the
system
for $ a $ and $ b $.

$ \left\{\begin{array}{l}3.00b + 3.70a = 105.00\\3.70b + 4.69a = 123.90\end{array}\right. $

(b)
Use
the
regression
feature
of
a
graphing
utility
to confirm
the
result
in
part
(a).

(c)
Use
the
graphing
utility
to
plot
the
data
and
graph
the linear
model
from
part
(a)
in
the
same
viewing
window.

(d)
Use
the
linear
model
from
part
(a)
to
predict
the demand
when
the
price
is $ \$1.75 $.

In question 77. We're making That's 500 equals the number of snow. No, and November is our starting great. So equals zero irritable. So I have plus 500 finds gives me home. So I have 500 plus 500 which is 1000 just that e equals January. Really fines sign. Oh, I over six times too, which is free and high over three co sign equals 1/2 by 500 waas by 100 Lying's one 750 just happened 500. We'll see. We're looking at February so pee or three plus five sign six three. So that's really quite over too. And value of sign that fly over to see Rose. So 510 00 So I only have 500 r d In May. We have p equals six. So by 500 plus 500 fine Oh, high over six times six. So just hi. Which equals negative one. So I have 500. What? E he was what, six Mom, which is really replace too at three point over, too. Sign zero Now they want me to do a graft with a free hand, right? I can't be 1000. Yeah,

Already. So we are looking to model the monthly sales of snow blowers. And so with the equation that's given, we can go ahead and find the anti derivative of it so that we can sum up all of the snow blowers that were found in january, plus all the snow blowers that were found in february All the way through December. And so instead of doing those calculations 12 times we can actually use the anti derivative to find the summation for us. So I've taken the equation from this problem and I split it into two because there's a plus side in the plus sign in the middle. And so let's go ahead and find the anti derivative. The anti derivative of 15 is going to be oops 15 T. And this will be evaluated from 1 to 12 because that's january to december. The second half of this equation is a little bit more challenging. So I'm gonna go ahead and rewrite this because we need to use U substitution because this thing here is kind of complicated. So I'm gonna go ahead and replace this with six sign of you D. T. Because I'm going to go over here and let you be equal to this whole thing, pi over six t minus eight. Now, if you notice here in the second line, I have a variable view but then I'm taking the derivative of A. T. And they don't match right now. So I need them to match. And so I'm going to come over here and take the derivative of both sides. And that should say to you. So the derivative of you would be Dutt and then the derivative of this guy will be pie over six. I need to get DT by by itself so that I can replace it in the equation. So I'm going to get D. T. is equal to six over pie, do you? And then I can come over here and again just bringing down the 15 for now. Alright so now my D. U. Matches the variable for this problem and we can go ahead and solve going to bring the six over pie out front and now The six is going to come down. The derivative of sine is negative co sign. So it's become a -6 co sign of you. And this is gonna be evaluated from 1 to 12. But those one and 12 a representative of the T. Variable. And right now we still have our variable in terms of you. So I have to make one more step to replace this you with what we labeled it up here. Now I can use the first fundamental theorem of calculus to plug in 12 and then plugging one and subtract the two. And that should give us the answer of how many snow blowers were sold during the course of the year. So let's go ahead and try that. so first up is the 15 I'm gonna plug in 15 times 12 -15 times one. And then eight of 6. Cosign of Pi over six. So T -8 is 12. -8. -8 is four. I'm going to subtract but then there's a negative. So I'm gonna make that double negative into a positive six. Co sign of pi over six. And then t minus 81 minus eight is a negative seven. Mhm. So, I'm just gonna rewrite this. I have seven pi over six Already, so 15 times 12 is 1 80 minus 15 is 165 plus six over pie. All right, let's go ahead and use our unit circle. We have native six and then co sign of four pi or six. Which is the same thing as co sign of two pi over three and we get negative one half and then plus six. Co sign of -7 pi over six is negative Radical 3/2. Go ahead and simplify that negative six times a negative one half is a positive three plus A- three Radical 3. Continuing to do this algebra. So we get six times 3 is 18 over a pie and then -18 radical three over pie. So I'm gonna use my calculator to solve the rest of this and let's see what we get When I plugged it into the calculator. I got 160 0.81 And if you would like to double check your answer after doing this much algebra, I highly recommend that you can go ahead and put this equation right into your calculator, make sure your window is formatted correctly and then go to second cowpoke. Um for me it's choice seven, it's the anti derivative. It will ask you what the lower bound is. The answer is one. The upper bound, the answer is 12 And you can confirm that the area under the curve between one and 12 is indeed 160 and some change. Additionally you can go to your table on your calculator and add up all of the amounts between one and 12. I'm just going to plan on doing that in my head just roughly to make sure that makes the most sense and it looks like most of these are around 10. Yeah it's like At one it's 18 to is 15, 3 is 12, four is 9.85 is nine. And then those numbers repeat going back up. So 160 I would say is makes sense to me. And so I'm going to go ahead and call that my final answer to Part A This same process is going to be used for part B except it's only asking for july through december. So instead of months one through 12, it's going to be months seven through 12. So I can actually come up here and change all of these ones two sevens, This should be a seven and then um This was evaluated at seven. So instead of 7 -8 or excuse me, instead of 1 -8 to be a negative seven, it's 7 -8 to be this should be a negative one. Okay already. So instead of doing the whole problem over again, I'm going to erase this bottom part and let's go from here already. So five times 12 was 1 80 minus Excuse me. 15 times 12 was 1 80 then 15 times seven is one of five. So that is a 75 plus six pi. This was co sign of three pi Over to excuse me again. This was co sign of two pi over three. This was a -6. So then times are negative one half. All together. That is a positive three plus six times co sign of negative pi over six. That is going to be a positive three radical 3/2. And so that is a positive 3/3. 3 RADICAL three. All right, let's go ahead and finish the rest of this on the calculator, 75 Plus 18 over pie. Yeah. And solving us on a calculator, I get 90.65. So that's how many um the snow blowers. Snow blowers were sold between july through december.

So in this problem we are given an equation that demonstrates the monthly sales of snow blowers. And this question in part is asking how many snowblowers are sold during an entire year. So we need to add up all of the snow blowers sold in january and february and March and april all the way through december. So the best way to do that would be to find the anti derivative aka the area under the curve, aka the sun. So we're going to go ahead and evaluate this equation. I split it into two anti derivatives because there's a plus sign between the 15 and then the trick function. And then we're going to evaluate this anti derivative using the first fundamental theorem of calculus between january and december. So let's go ahead and get started. The anti derivative of 15 Is just 15 groups. They don't actually need to write this symbol anymore is 15 T. And like I said, that's going to be evaluated between zero and 12, which they are telling us in the problem. Plus now when it comes to this trade function, um this is sign of this whole thing in here. And so if we were to take the derivative of this, it would require chain rules. So now when we're taking the anti derivative we need to use U. Substitution because we know what the anti derivative of sine of X. Is. But it gets a lot more complicated when you have signs of two X. Or in this case sine of pi over six times the quantity X minus eight. And so in order to deal with that more complicated um step we go ahead and use what we call U substitution. So I'm gonna come over here and and make that long thing equal to you So I can come down here and rewrite this. This is now six sign of you D. T. So I've taken care of the fact that now I know what the anti derivative of sine of you is, but I'm still multiplying by D. T. And so my variables at the moment don't match. So I need to rearrange some things over on the right here to get D. T. In terms of the you. Instead I need my variables to match. So I'm gonna go ahead and take the derivative of you, which is do you over D. T. And then this is really Hi over six T minus eight, pi over six. So the derivative of pi over 60 is just pi over six. And then the during a pi over six, that's a constant term. So the derivative of that is zero. Like I said, I want to replace the DT with A D. U. So I need a equation that starts with DT equals. so I can do a direct substitution rearranging this equation up here. I can cross multiply and I get DT is equal 2 6 over pie. Do you? So on this line I'm gonna be rewriting this 15 TA couple more times D. T. Is the same thing as six over pi D. You. So I can do a direct replacement here. And now I have my trig in terms of something I know how to take the anti derivative of and I have matching variables. So I can go ahead and do the anti derivative. Now I'm going to bring the coefficient out in front until six times six over pie is 36 over pie. The anti derivative of sine is negative co sign. So I'm going to make that a negative And again this is being evaluated between zero and 12. Now that zero and the 12 is in terms of T. It is not in terms of you. So, we need to plug this value back in now. Everything is matching as it should be. Let's go ahead and finish the algebra. I'm going to use the first fundamental theorem of calculus, which means I plug in the larger number in this case 12 and then subtract off the smaller number. So we can do 15 times 12 minus 15 time zero minus. Oops. I'm gonna write that 36 over pie outside. All right. So, let's do some mental math here plugging in 12 for T. So 12 minus eight is four, that is four pi over six. Which is the same thing as two Pi over three. And then I can go ahead and plug it in again, plugging in 00 -8 is negative eight over six. So that is co sign of negative 8/6 which is the same thing as for over three. Okay 15 times 12. 180 plus 36 over a pie. And then coastline of 2/3 Using the unit circle is negative 1/2 Minour coastline of negative 4/3 Is also negative 1/2. So these two end up cancelling each other out. Now you have one half plus one half becomes zero. You are multiplied by anything is just zero. So the answer Is 180. So that was the answer for part a asking for the total monthly sales in an entire year. Part B is doing something similar but just july through december. So we can scroll down here and instead of redoing the whole thing we can take this line And instead of evaluating from 0 to 12 we can evaluate it from 7 to 12. So let me rewrite this 15 T evaluated from 7 to 12 minus 36 pie over five co sign of Pi over six. Two minus eight Evaluated from 7 to 12. So using the first fundamental theorem of calculus again we're going to do 15 times 12 -15 times seven. I put that back, it's just too late. Make it super clear minus. Okay, so as I'm writing this, I'm actually realizing that I had one too many minus signs up here so I can go ahead and take one of these out. My apologies and then this should be a positive a negative but it didn't end up mattering because things ended up canceling out here to zero. So it did not affect the answer but I'm going to get it right this time in the green so minus 36 over pie. And then inside the brackets we've got co sign of. Okay, so 12 minus eight is 44 pi over six is two pi over three minus both. Sign Of 7 -8 is a -1. So negative pi over six. Mhm. This was 1 80 15 times 12 was 1 80 15 times seven Is 105. So we get the difference here to be 75 -36 over pie. Co sign of two pi was To buy or three was negative 1/2 and co sign of negative pi over six is Radical 3/2. Yeah. All right. So I'm gonna go to my calculator to solve the rest of this and I plugged all of this in and I got 90 0.65 And so let's just double check and make sure that makes sense. In the green we are only talking about july through december, whereas in the purple, we were talking about the entire year. So july through december should be less than the purple. And it is, it actually looks like it's cut right in half, which is awesome. So if someone is talking about monthly sales or something like that in context, it sounds like they're being pretty consistent in terms of one half of the year and the other. You can also double check this using your calculator. I actually walk through that in problem. 56, much more in depth problem. 56 uses a graphing calculator and does not do the algebra. So check that problem out. If you would like to see how to double check this on your calculator.

Hello. So we have this problem where a store manager wants to know the demand for your product as a function of the price. And we're given this table where we have different prices that are represented by X. And we have the demand which is why or the daily sales. So for part A we have to find the least squares regression line by solving the system given to us. So let's start doing that right? Hey we have three B Plus 3.7 A. She goes to 105. Now mind you This 3.00 is the same as three at 3.70 3.7. I just made it simpler to write so we're going to use this first equation and saw for a now to do that first we're gonna is attracted to be on both sides. We'll get 105 -3 B. She goes to 3.7 a. Then we're gonna divide both sides by 3.7 Which is a equals 105 -3 b. Over 3.7. Now we know that a decimal can't be on the bottom but we're just going to leave it there for now and handle it a little later. So we're gonna use a and plug it into the second equation. So we have 3.7 b Plus 4.69. Times 105 minus three B. And this is over 3.7. Chief was to Laundry 23.9. Now We're going to distribute the four 69 into the parentheses to do so we're going to transform the decimals into fractions. So we have 416 9/1 100. I mean times that By 105 -3 B. This is gonna be over 37. Over Oh over 10. Sorry now notice how there's a fraction on the bottle. So we're just gonna do keep change flip meaning we're going to keep the numerator, changed the division to multiplication and then flip this fraction So it's going to be 105 -3 b. Times 10. And this is going to be divided by 37. Now bring down 469 Over 100. Well we know that 10 divided by 100. We just we could just do this And we're left with 10 on the bottom here Now 10 times 37 is 370. And then on top we're going to have 469 times 105 minus three B. And just so we remember let's bring down our other terms. 1:23.9 and then our three point seven B. Yeah. Let's see if we distribute 469 and the parentheses we'll get 49245 -4, 1807 or 1407 B. And this is over 3 70 And we got 3.7 b. And then the 123.9, that's now that's settled. We're going to Transform this 3.7 B by multiplying it by 370 over 370. So we could combine it with this numerator. Once we do that, we're going to end up with 1369 Be over at 3 70. Mhm. And a bad deal. The terms 49245 1407 B. Shaw equals to 123.9. Now We're going to multiply both sides by 370. These two cancel 3 70 here. So This time is 3 70 is 45,843. We write everything in the numerator, So 49002 RJ 45 five minus 1004 07 B. Yeah, notice how there's a B. Here and here. So we're going to combine these two together and we're going to end up with mm hmm, -38 B. Bring down the 49245 And set that equal to 45,843. Now we're gonna subtract both sides by 49245, 9245. And then we'll end up with a negative 38 B, which is equal two, -3402. We're going to divide both sides by negative 38 and we're gonna get a long decimal but Let's see is 89 50 two then six. And we're gonna round. So this of course this We're gonna round up, so this is approximately 89 53. Now that we got B we're gonna plug it in to the other equation to get A. So we have three times 89 0.53 Plus 3.7 a. Which equals to 105. and then three times 89.53 is 268.59. Bring down the 3.7 A 105. We're going to subtract 268.59 on both sides, 68.59. Bring out the 3.7 a. Now the difference between these two is negative, 100 is 63.59, Divide both sides by 3.7 And you're left with a equals negative 44.21. Now our regression line equation is Y equals -400. I mean 44 0.21 X plus 89 0.53. Now for part B of this problem, we have to see if this checks out. So I went ahead and plugged in the values and the from the chart into dez most. And let's see, we got M equals negative 44.8 21 05. Which rounds down to our um or a. And then we got B. Which is 89 .5263. Which we rounded to 89.53. So our answer holds up. So now for part see we have to use our model to predict it. The man when the prices $1.75. So are we going to do is just plug it in S. X. Into our equation. So we can have Y equals negative hold for 0.21 times $1.75 plus 89 0.53. These two multiplied together were equal to -77.36 75, and then Plus 89 2053. And we get y equals 12.1625. Of course, this is not a real dollar amount. So our answer would be why, I mean our demand, It would be about 12.16, Or because on the chart it's rounded to whole numbers, so we're just gonna round it down to 12. So that's our demand.


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