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July 22nd, 2009, 03:28 PM   #1
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Algebra Applications

Hey everybody, I'm going to level with you here, I pretty much slipped through the cracks in highschool without learning much of anything and went to work in a factory. Now I'm old but trying to get my act together academically. I'm going to post 8 questions below, the answers for which I could maybe guess at, but it is imperative that I answer them correctly and I have no way of reasonably checking my answers. I know that you probably get a lot of this sort of thing and most of you probably couldn't care less that I'm here, but if someone wants to help me out by answering some of these and maybe showing their work so that I can figure out what's going on, then I would greatly appreciate it. If you ever end up owning a factory, I'll reciprocate by showing up and totally doing the mind numbing labor for you.

Thanks,

1. Chuck, a quality control engineer at Ricky's Reasonable Rockers, Inc., found that 3.2% of the rockers in a recent shipment from his factory were defective due to a flaw in the wood. If Chuck discovered that 8 rockers in the shipment were defective, how may rockers were in the total shipment?

2. At Ricky's Reasonable Rockers, Inc., owner, Ricky Ray, purchased new sanding equipment for his rocker production line at a cost of $15,000. If the sanding equipment depreciates at $1,600 per year, find the following:
A. Find a value function, V(t), expressed as a function of time (t) in years.
B. Find the value of the equipment after 4 years and 3 months.
C. If Ricky Ray wants to replace the equipment when the value of the equipment is $5,000, how many years will it be before he has to replace the equipment?

3. In April, Ricky's Reasonable Rockers, Inc., purchased 8 work shirts and 17 pair of work pants for its production line workers at a cost of $351. Then again, in August they purchased an additional 16 work shirts and 13 pairs of work pants for $387.
A. How much was each work shirt and each pair of work pants?
B. Next year Ricky's Reasonable Rockers, Inc. plans to buy 40 work shirts and 25 pairs of work pants. If they have budgeted $750 next year will they have enough funds to purchase the shirts and pants?

4. Ricky's Reasonable Rockers, Inc. makes and sells wooden rockers. (the hell you say). The daily revenue from the sale of those rockers is given by the function, R(x) = x ^2 - 18x. The owner, Ricky , has determined that the daily cost of producing his rockers is described by the function, C(x)=81+6x. While many think that Ricky is off his rocker with his low, low, prices; how many rockers must Ricky make and sell daily to break even?

5. The cost of removing a percentage, p , of the air pollution from the stack emissions from sanding and lathing at Ricky's Reasonable Rockers, Inc. is modeled by the following function: C(p)=130000p/1-p
A. What percent of the air pollution can Ricky's Reasonable Rockers, Inc. remove if they are willing to spend $550,000?
B. If the USEPA requires Ricky's Reasonable Rockers, Inc. to remove at least 95% of their pollution, how much will Ricky's Reasonable Rockers, Inc. have to spend on air pollution control?
C. Can Ricky's Reasonable Rockers, Inc. remove 100% of their air pollution? Why?

6. At Ricky's Reasonable Rockers, Inc., Frank, the maintenance man, needs to fix a leak in a pipe on the air pollution control equipment at the factory. Frank has a ladder that is 20 feet long and he knows that according to the OSHA safety manual that to make a safe climb he must have the base of the ladder at least 8 feet from the base of the building. If the pipe on the building that he wants to repair is 24 feet high, does the ladder reach the pipe so Frank can safely make the repair? (I think this would use the pythagorean theorem but I'm not sure)

7. Frustrated with dealing with the USEPA about the air pollution at his plant, Ricky took his cell phone and threw it off the top of his office building. (obviously not an AL Gore, man.) The height h(t) of the cell phone at time (t) is described by the function: h(t)= - 16t^2+64t+192
A. When will Ricky's cell phone be at a height of 240 feet?
B. How many seconds will it take before Ricky's cell phone hits the ground?

8. Ricky is thinking about selling his business. His original investment was $243,000 and he'd sell if he can get at least half a million dollars (you'll enjoy rocker production, it's a growth industry). The value of his business has appreciated at an average of 6% for the 12 years since he started. Based on this appreciate rate (assuming interest compounded annually), would Ricky sell his business? Also, What is the value of the business had it appreciated at an average of 7%?
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July 22nd, 2009, 04:49 PM   #2
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1. Since "%" means "divided by 100", if there were k rockers in total, (3.2/100)k = 8. Hence k = 800/3.2 = 250.

2A. V(t) = $15,000 - $1,600t.
  B. V(4.25) = $15,000 - $1,600(4.25) = $15,000 - $6,800 = $8,200.
  C. V(t) = $15,000 - $1,600t = $5,000, so t = ($15,000 - $5,000)/$1,600 = 6.25 (which corresponds to 6 years 3 months).

3A. If, in August, the prices were the same as in April, 16 shirts and 34 pairs of work pants would have cost $702. The actual cost was $315 less, corresponding to buying 21 pairs of work pants fewer, so the price for each pair of work pants was $315/21, i.e., $15, and so the 13 pairs bought cost $195. Hence the 16 shirts cost $192 and were $12 each.
  B. At the same prices, 40 work shirts and 25 pairs of work pants would cost $480 + $375 = $855, which would be $105 over the budget.

4. Breaking even implies R(x) = C(x), so x² - 18x = 81 + 6x, i.e., x² - 24x - 81 = 0, i.e., (x + 3)(x - 27) = 0. Hence if x is the number of rockers sold daily, x = 27.

Can you make an attempt at the rest? I'd like to see how you get on.
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July 22nd, 2009, 05:53 PM   #3
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Re: Algebra Applications

I really do appreciate your help. having the answers is really giving me big helping hand for figuring out how to wrap my mind around these things, I really don't know why anyone would give out problems without including the answers for students short of a test situation, but there ya go.

anyhow as per the others, I'm thinking along these lines.

5. A) I'm pretty much at a loss here, since the amount they're willing to spend is 550,000 it should probably be substituted for C, so 550,000(p)=130,000(p)/1-p then I go kinda haywire because 130,000 divided by 1 is 130,000 and to my (probably erroneous) way of thinking a p divided by a -p would be -p. so then 550,000(p)=130,000-p my inclination then is to add a p to both sides and subtract 550,000, which gives me a pretty ridiculous answer.

B) so here they have to remove 95% of their pollution which gives me (p) so I have C(.95)=130,000(.95)/1-(.95) I then went ahead and solved for the numerator and denominator seperately and got 123,500/0.05 which after division then becomes 2,470,000. So I have C(.95)=2,470,000 I then removed the .95 from the C by dividing both sides by .95. I finished with C=2,600,000

C) This one is doing a good job of proving to me that I don't know whats going on because because when I try to solve for 100% I determine that it is 1.00, so my equation turns into C(1)=130,000(1)/1-(1) which essentially is telling me that to remove 100% of his factory's pollution, Ricky only needs to spend the 130,000 given in the equation.

I will go ahead and post this while I work up the others so that everyone can have a good chuckle at my expense while I slough through these others.
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July 22nd, 2009, 06:21 PM   #4
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Re: Algebra Applications

Ok so the others

6. I don't actually feel to bad about this one (further proof that ignorance is bliss) so here's what I've got
This problem is essential a square, between the shape of the ladder and the pipe and wall. so using the pythagrean theorem I ended up with 24^2 = 20^2 +8^2, or 576= 400 + 64 , making my final answer 576=464, now since the pipe represents the C^2 of theorem then that would mean that he could not reach it, because the C^2 is the higher number.

7. this one is totally losing me, seems more like physics to me, which if you can believe it I have less experience with than simple algebra. Anyhow heres what I did
A) h(t)= - 16t^2 +64t +192, I substituted in 240 from the question for h. so then I have 240(t)= - 16t^2 +64t +192, then I pretty much stopped. I kind of want to subtract the 192 from both sides but I assume that I can't because the other side is actually 240t. That being the case I subtracted the 64t instead, leaving me with 176t= -16t^2 +192, then Ideally I want to divide both sides by 176 but that gives me some ridiculous decimal which seems very wrong, even if it didn't I wouldn't be sure what to do with the remaining -16t^2

B) is currently a very similar scrawling of nonsense

8.I'm not sure if this is right but the closest relevant formula I could find in my math book was A=P [1+r/n]^nt I decided to use it and this is what I got. 243,000[1+.06/12]^12 x 12 which became 243,000(1.005)^144 = $498332.45 which means that Ricky is going to be building rockers for the rest of his life.
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July 23rd, 2009, 03:47 PM   #5
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5. I think the intended meaning was as writing C(p) in defining C as a function is just a (redundant) way of indicating that the value of C depends on the value of p; C(p) in that context doesn't denote C multiplied by p. My interpretation for the right-hand side of the equation is largely because any alternative wouldn't make much sense. It should also be assumed that the cost is $C.

A) Putting 550,000 for C in the equation gives 500,000 = 130,000 p/(1 - p), i.e., 50(1 - p) = 13p.
    Hence 50 - 50p = 13p, and so p = 50/63 = (5000/63)%. That's roughly 79%.

B) Putting p = 95/100 = 0.95 in the equation gives C = 130,000(0.95/0.05) = 130,000(19). So the amount that will need to be spent is $2,470,000.

C) Putting p = 1 in the equation gives C = 130,000/(1 - 1) = 130,000/0. Since division by zero is undefined, the equation can't be applied in this case, but it would be reasonable to say that 100% of the pollution can't be removed, since the value of C given by the equation grows without bound as p approaches 100% and the company presumably doesn't have unlimited funding.

6. The question says the pipe is 24 feet high. The ladder reaches only 20 feet even if (unsafely) placed vertical, and I certainly wouldn't like to fix a pipe that is above my head even when I'm standing on the top rung of a ladder. Although Frank could probably reach slightly (perhaps 3 or 4 feet if he is tall) above the top of the ladder, the safely positioned ladder's reach is given by the Pythagorean theorem as ?(20² - 8²) feet, which is about 18 feet 4 inches, about 5 feet 8 inches short of the pipe.

7. Use h = -16t² + 64t + 192, where h feet is the height after t seconds.

A) To solve 240 = -16t² + 64t + 192, write it as 16t² - 64t + 48 = 0, i.e., 16(t - 1)(t - 3) = 0. This indicates the cell phone will be at a height of 240 feet after 1 second and after 3 seconds.

B) To find when the cell phone hits the ground, find the positive value of t for which -16t² + 64t + 192 = 0. The equation can be written as -16(t + 2)(t - 6) = 0, so t = 6.

8. For annual (as distinct from monthly) compounding, the value after 12 years is given by $243,000(1 + .06)^12, which is about $488,963.74, so Ricky would be able to sell if he waits a few months. He can sell now if the appreciation rate averaged 7% (you should be able to do the calculation).
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