Question 662256
{{{x}}} = number of quarts
{{{y}}} = price in $
 
(Explanation: We have to start by defining variables that will be related by a linear equation. Next, we have to use the two (x,y) data points given to find the equation)
 
one gallon (4 quarts) for $3.09 --> (4,3.09) (x=4, y=3.09)
half gallon (2 quarts) for $1.65 --> (2,1.65) (x=2, y=1.65)
 
a) Write the particular equation expressing price in terms of quarts.
(Explanation: We are looking for an equation of the form {{{y=mx+b}}}. We can substitute the (x,y) values for each data point to get a system of equation to solve for {{{m}}} and {{{b}}}. Otherwise, we can use the two points to calculate the slope {{{m}}} of the line, and then we can find the rest of the equation one way or another)

Using a system of equations:
{{{system(3.09=4m+b,1.65=2m+b)}}}
Subtracting the second equation from the first equation;
 {{{3.09=4m+b}}}
-{{{1.65=2m+b}}}
-----------------
 {{{1.44=2m}}} --> {{{1.44/2=2m/2}}} --> {{{highlight(m=0.72)}}}
Substituting that value in the second equation, we get
{{{1.65=2*0.72+b}}} --> {{{1.65=1.44+b}}} --> {{{1.65-1.44=1.44+b-1.44}}} --> {{{highlight(0.21=b)}}}
The equation is {{{highlight(y=0.72x+0.21)}}}

Using slope:
(Explanation: the slope, {{{m}}} in the change in y divided by the change in x, as we go from one data point to the other.)
{{{m=(3.09-1.65)/(4-2)}}} --> {{{m=1.44/2}}} --> {{{highlight(m=0.72)}}}
The point-slope form of the equation, using point (2,1.65), is:
{{{y-1.65=0.72(x-2)}}}
{{{y-1.65=0.72(x-2)}}} --> {{{y-1.65=0.72x-1.44)}}} --> {{{y-1.65+1.65=0.72x-1.44+1.65)}}} --> {{{highlight(y=0.72x+0.21)}}}
 
b) If Handy Andy sold 3-gallon cartons, what would your equation predict the price to be?
{{{3gallons=3*4quarts=12quarts}}}
Substituting {{{x=12}}} in the equation, we get
{{{y=0.72*12+0.21}}} --> {{{y=8.64+0.21}}} --> {{{highlight(y=8.85)}}}
Andy would sell a 3-gallon carton for $8.85. 
 
c) The actual prices for pint cartons (1/2 quart) and one-quart cartons are $.57 and $.99, respectively. Do these prices fit your mathematical model? If not, are they higher than predicted or lower?
The predictions are found using the equation {{{y=0.72x+0.21}}}
For 1/2 quart = {{{0.5}}} quart, {{{x=0.5}}}, so
{{{y=0.72*0.5+0.21}}} --> {{{y=0.36+0.21}}} --> {{{highlight(y=0.57)}}}
For 1 quart, {{{x=1}}}, so
{{{y=0.72*1+0.21}}} --> {{{y=0.72+0.21}}} --> {{{highlight(y=0.93)}}}
the price for pint cartons agrees with the {{{y=0.57}}} prediction,
but the price for the quart, $0.99, is higher than the {{{y=0.93}}} prediction.
 
d) Suppose that you found cartons of milk marked at $3.45, 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?
I would use {{{y=3.45}}} and the equation {{{y=0.72x+0.21}}} to find the number of quarts, {{{x}}}.
{{{3.45=0.72x+0.21}}} --> {{{3.45-0.21=0.72x+0.21-0.21}}} --> {{{3.24=0.72x}}} --> {{{3.24/0.72=x}}} --> {{{highlight(x=4.5)}}}
Such a carton would hold 4.5 quarts according to the model.
 
e) What does the price-intercept represent in the real world?
The price-intercept is the y-intercept in the equation {{{y=0.72x+0.21}}}.
It is the value, {{{0.21}}}, representing $0.21, the price of a carton with zero quarts of milk inside. It represents the cost of the carton plus maybe Andy's work making a sale.
 
f) What are the units of the slope? What real-world quantity does this number represent?
The slope, {{{0.72}}}, represents the cost of the milk per quart.
The units are $/quart, and it gets multiplied by the number of quarts to find the price of the milk inside the carton in $.

g) Sketch and label the graph.
{{{drawing(300,300,-1,14,-1,9,
grid(1),
blue(line(0,0.21,14,10.29)),
blue(circle(0,0.21,0.2)),blue(circle(4,3.09,0.2))
)}}} To graph, we just need to plot 2 points (blue circles) and connect then with a line.
The blue line represents {{{y=0.72x+0.21}}}.
The blue circles represent
the intercept point, (0,0.21), {{{x=0}}} with {{{y=0.21}}}, and
point (4,3.09), {{{x=4}}} with {{{y=3.09}}}, representing 4 quarts at $3.09.