Two Algebra Word Problems
Date: 3/23/96 at 13:50:42 From: aweare Subject: grade 11 math problems Hi! A friend and I are stumped on these questions. Can you help? 1) Two people decide to open a music store. A building is available for rent for $6000 per year. Heat, electricity, taxes, etc... are estimated at $5000 per year. Each person wants to make $7 per hour, and will work 10 hours per day for 300 days of the year. If CDs cost 7.50 each, but will sell for $12 each, how many CDs do they have to sell? If they can sell 25 CDs per day for $12, and they can sell 10 more each time they decrease the price by $0.25, how much should they charge, and what would be the annual profit or loss? 2) The side of one cube is 2cm more than the side of another cube. The volumes of the cubes differ by 100cm cubed. Find the length of the sides of each cube correct to two decimal places.
Date: 5/17/96 at 15:35:54 From: Doctor Ethan Subject: Re: 4 grade 11 math problems Okay, great questions. >1) Two people decide to open a music store... Okay so we have two questions: 1. How many CDs must we sell under these conditions? 2. How do we maximize our earnings under different conditions? 1. Our expenses total $11000 and we want to make 2 * 300 * 10 * 7 = 42000 So the total is 53000 And for each CD you sell you make $4.50 So 53000/4.5 = 11778 CDs in a year, which is 11778/300 = 39 CDs in a day (assuming the you are going to be open 300 days a year) or 11778/365 = 32 Now for question 2. I am going to assume that both of you are working at all times that the store is open - I am assuming that it is open 10 hours a day for 300 days a year. At $12 a day you will sell 25 CDs, make $4.50 on each CD, and have net earnings of 4.5 * 25 * 300 = 33750 so you lose 53000 - 33750 = 19250 (assuming the same desired salary.) Now let's assume we decrease the price by $.25. Then our earnings are 4.25 * 35 * 300 = 44625 (so we are doing better) Rather than continue testing each case, let's write down a formula that will help us. Let x be the amount in quarters that we lower the price from $12 so the original case is the x = 0 case. Now the equation looks like this. (4.5 - x * .25) * (25 + 10x)* (300) = 300(112.5 + 38.75 x - 2.5 x^2) (The earning per CD) * (the # of CD's) * (days) And that = 33750 + 11625 x - 750 x^2 And you want to know when this function achieves a maximum. I will let you figure it out. Either graph it or use calculus. .............................................. >2) The side of one cube is 2cm more than the side of another cube... Call the length of the big cube y and the little cube x So y = x + 2 and x ^3 + 100 = y^ 3 So 100 + x^3 = (x+2)^3 Now see if you can solve this for some positive number x. -Doctor Ethan, The Math Forum
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