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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 

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 

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


x ^3 + 100 = y^ 3


100 + x^3 = (x+2)^3

Now see if you can solve this for some positive number x.

-Doctor Ethan,  The Math Forum

Associated Topics:
High School Basic Algebra

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