Word problem, price of milk

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doc74

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Hi everyone, I have worked on this for the longest time, but it's just not clicking.

I posted the entire problem but would be happy with just the coordinates and how to get there. Any help is greatly appreciated!

The current average price for one-gallon cartons (4 quarts) of milk is $3.22 and the price of half-gallon cartons is $1.71. Assume that the number of cents you pay for a carton of milk is a function of the number of quarts the carton holds.

Write the equation that expresses price in terms of quarts.

If the local store sold 3-gallon cartons, what would your equation predict the price to be?

The actual price for pint cartons and one-quart cartons are $.59 and $1.03, respectively. Do these prices fit your mathematical model? If not, are they higher than predicted, or lower?

Suppose that you found cartons of milk marked at $3.60, but that there was nothing on the carton to tell what size it is. According to your model, how much would such a carton hold?

What does the price-intercept represent in the real world?

What are the units of the slope? What real-world quantity does this represent?
 
doc74 said:
Assume that the number of cents you pay for a carton of milk is a function of the number of quarts the carton holds.

What kind of function ?

Linear, quadratic, radical, logarithmic, something else ?
Click to expand...
 
Now write a linear function.

You are given two points (c,q) as (322,4) and (171,2) - draw a straight line through the points, and find the function.

The equation of a line passing through (x[sub:3rh6gi3q]1[/sub:3rh6gi3q],y[sub:3rh6gi3q]1[/sub:3rh6gi3q]) and (x[sub:3rh6gi3q]2[/sub:3rh6gi3q],y[sub:3rh6gi3q]2[/sub:3rh6gi3q]) is:

y−y1y2−y1 = x−x1x2−x1\displaystyle \frac{y - y_1}{y_2 - y_1} \ = \ \frac{x - x_1}{x_2 - x_1}

and now continue....
 
Subhotosh Khan said:
Now write a linear function.

You are given two points (c,q) as (322,4) and (171,2) - draw a straight line through the points, and find the function.

The equation of a line passing through (x[sub:cg2b32qg]1[/sub:cg2b32qg],y[sub:cg2b32qg]1[/sub:cg2b32qg]) and (x[sub:cg2b32qg]2[/sub:cg2b32qg],y[sub:cg2b32qg]2[/sub:cg2b32qg]) is:

y−y1y2−y1 = x−x1x2−x1\displaystyle \frac{y - y_1}{y_2 - y_1} \ = \ \frac{x - x_1}{x_2 - x_1}

and now continue....
Click to expand...

First of all, thank you for your help, I do appreciate it.

Hmm looks like I turned the coordinates around...so with those coordinates my slope would be 2/151.

Using one set of points that gives me y=2/151x - 40/151

Somehow that seems wrong... what am I missing..?
 
It would appear you have your x and y switched.

The points are (4, 3.22) and (2, 1.71)

m=3.22−1.714−2=1.512\displaystyle m=\frac{3.22-1.71}{4-2}=\frac{1.51}{2}

Now use either set of points for x and y to plug into y=mx+b and solve for b.

Then, for 3 gallon, use x=12.

For the other part, set your line equation equal to 3.6 and solve for x (number of quarts).
 
> Using one set of points that gives me y=2/151x - 40/151

That's correct; which means x = (151y + 40) / 2

Since y is the number of quarts, yer all set; like how much for a 9quart container:
x = (151(9) + 40) / 2 = 699.5 cents or 7 bucks rounded...ya'll ok now?
 
Awesome people! Thank you so much.

So let's see if I'm doing this right...

f(y) = x

(4,3.22) (2,1.71)

y=2/151x - 40/151

x = (151y + 40) / 2 x = price in cents, y = number of quarts


3 gallons = 12 quarts so y = 12

x = (151(12) + 40) / 2

3 gallons = 926 cents or $9.26

The actual price for pint cartons ( quart) and one-quart cartons are $.59 and $1.03, respectively. Do these prices fit your mathematical model? If not, are they higher than predicted, or lower?

Pint carton: 1 quart price = 115 cents or $1.15

59 = (151(0.5) + 40) / 2 price does not match and should be 57.75 cents

Quart carton: 1 quart price = $1.03


103 = (151(1) + 40) / 2 price does not match and should be 95.5 cents

Both prices are higher.

Suppose that you found cartons of milk marked at $3.60, but that there was nothing on the carton to tell what size it is. According to your model, how much would such a carton hold?


360=(151y+40)/2

(2)360=151y+40

720=151y+40

680=151y

y=680/151

y=4.5 quarts

What does the price-intercept represent in the real world? no clue...

What are the units of the slope? What real-world quantity does this represent? again no clue...
 
doc74 said:
f(y) = x

(4,3.22) (2,1.71) These are (y, x) pairs. Unusual, but okay.

y=2/151x - 40/151

x = (151y + 40) / 2 x = price in cents, y = number of quarts

This is okay, if you're sure that they want a function that generates the price in cents, instead of in dollars. The units on the two given prices are dollars.

I mean, the given coordinates are not (4, 322) and (2, 171).

I got the price function p(q) = 0.755q + 0.2

p(q) = average carton-price (in dollars) and q = carton volume (in quarts)

I would graph my function using a qp-coordinate system. (KewPee Koordinates, heh, heh.) That is, the horizontal axis on my graph would be called the q-axis, and the verical axis would be the p-axis.

My ordered pairs are (q, p) coordinates, representing price (p) as a function of volume (q). In ordered pairs, the independent variable is always listed first and the dependent variable is always listed second.

You would graph your function using a yx-coordinate system. That is, your horizontal axis would be the y-axis, and your vertical axis would be the x-axis. That's because you chose the symbol y to represent the independent variable, so your x is a function of y.

Unusual, but okay. (These letters are just names, but conventionally we order the symbols as (x,y) pairs instead of (y,x) pairs, so we conventionally write y =f(x) instead of x = f(y). But hey, why follow convention? Blow your own horn! Just be sure to include your symbol definitions above because people will be confused.)

3 gallons = 12 quarts so y = 12

x = (151(12) + 40) / 2

3 gallons = 926 cents or $9.26 This is correct.

The actual price for pint cartons ( quart) and one-quart cartons are $.59 and $1.03, respectively. Do these prices fit your mathematical model? If not, are they higher than predicted, or lower?

Pint carton: 1 quart price = 115 cents or $1.15 I got 1.155, and that's $1.16 rounded.

But this is the quart unit-price for pint cartons. Why are you reporting it ?

59 = (151(0.5) + 40) / 2 price does not match and should be 57.75 cents How does one pay 0.75 cents?

Are you telling me that you take a hacksaw to your coins?

I got p(0.5) = 0.58, so the actual price is one cent higher.

Quart carton: 1 quart price = $1.03

103 = (151(1) + 40) / 2 price does not match and should be 95.5 cents How does one pay 0.5 cents?

I mean, if you're telling me that you have some U.S. half-cent coins, I will buy all of them from you. I'll even pay you 1¢ each; that's twice their "value"!!

I got p(1) = 0.96, so the actual price is seven cents higher.

Both prices are higher. This is correct.

Suppose that you found cartons of milk marked at $3.60, but that there was nothing on the carton to tell what size it is. According to your model, how much would such a carton hold?

360=(151y+40)/2

(2)360=151y+40

720=151y+40

680=151y

y=680/151

y=4.5 quarts This is correct.

What does the price-intercept represent in the real world? no clue...

I'm thinking that it's the average fixed cost associated with bringing one carton of milk to the market, regardless of volume.

The p-intercept (your x-intercept, oddly enough) is 0.2, so that's 20 cents added to the carton-price to cover the fixed cost of everything but the milk itself.

The cost of the milk itself comes from the 0.755q term, in the linear polynomial that defines p(q).

What are the units of the slope? I say that they are $/quart, but you say that they are ¢/quart.

What real-world quantity does this represent? m = 0.755 $/quart

This is the averaged price per quart (in dollars) of the milk only (i.e., not including the fixed "non-milk" costs that add 20 cents per carton).

This quantity is also the rate at which the average carton-price of milk increases (as a function of volume). ALL slopes are rates.

Note that the rate at which the carton price increases is not affected by the fixed costs. If the fixed costs were 40 cents, instead, then the p-intercept would be 0.4, but the slope (i.e., the rate of change) would remain the same. Graphically speaking, this means the line would simply be shifted up (vertically) 0.2 units, but the "slant" would remain the same.
Click to expand...
 
Thank you so much for that explanation, this is helping me a ton! I do not take a hacksaw to coins no, I have plenty dumb hobbies as it is! But I see your point, it asks for cents, so give cents.

Thanks everyone for your help, I think this stuff is finally sinking in. I'm going to practice some more!
 
doc74 said:
...... I think this stuff is finally sinking in. I'm going to practice some more!
Click to expand...
QUIT the humility; you're smarter than you're letting on :idea:
 
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