Question 202700: Suppose you have a lemonade stand, and when you charge $2 per cup of lemonade you sell 120 cups. But when you raise your price to $3 you only sell 60 cups.
a. Write an equation for the number of cups you sell as a function of the price you charge.
b. Denote "C" for number of cups, and "P" for the price you charge.
c. Assume the function is linear.
Continuing our lemonade stand question:
a. We all know that total revenue (TR) is a function of the price we charged (P) multiplied by the item quantity sold (in our case – Cups), i.e., TR = Price * Cups
b. Please write the equation for your TR by inputting your answer from the function you have calculated in question #2.
c. What price would maximize your TR?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you can graph this as a linear equation by taking the y value and making it the price and taking the x value and making it the quantity of cups sold.
for the equation I will be using y = m*x + b which I will be substituting and changing to c = m*p + b down below after all the work has been done.
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formula for a straight line is:
y = m*x + b
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m = slope = delta y / delta x = (y2 - y1) / (x2 - x1)
you have two points.
first point:
when price is 2, quantity sold is 120
when price is 3, quantity sold is 60
x is the price
y is the quantity sold
x1 = 2, y1 = 120
x2 = 3, y2 = 60
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(y2-y1)/(x2-x1) = (60-120)/(3-2) = -60/1 = -60
your slope = m = (-60)
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you know the slope; now you want to find the y intercept.
take any point on the graph that you know.
(x1,y1) should do.
(x1,y1) = (2,120) from above.
plug it into the equation y = m*x + b to get:
120 = -60*2 + b
solve for b
120 = -120 + b
add 120 to both sides to get:
b = 240
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you now have the equation for the straight line.
m = -60
y intercept = 240
equation is:
y = -60*x + 240
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you test this to be true by substituting known values and look for the answer you expect.
when y = 2, x = 120
in the equation, y = -60*x + 240, substitute 120 for y and 2 for x to get:
120 = -60*(2) + 240
which becomes
120 = -120 + 240
which becomes
120 = 120
which is true so the equation is good.
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your equation is good, but they wanted you to let c = number of cups sold and p = price, so you need to substitute c for y and p for x to get:
c = -60*p + 240
that's your equation.
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Total Revenue is a function of price * quantity.
our linear formula calculated quantity as a function of price.
If Revenue is to be calculated, we can used this formula (I think).
Our formula for quantity sold is:
c = -60*p + 240
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we should be able to use this same formula to solve for price as follows:
c - 240 = -60*p
p = -(c-240)/60 = (240-c)/60
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we now have a formula for c, and a formula for p.
let r = total revenue = price * quantity
since price = p and quantity = c, we have:
r = p*c
we know that c = -60p + 240
we also know that p = (240-c)/60
our formula becomes:
r = p*c = (-60p+240)*(240-c)/60
we multiply both sides of this equation by 60 to get:
60*r = (-60p+240)*(240-c)/60
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we solve this eqauation to get:
60*r = -14400p + 60p*c + 57600 - 240c
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we divide both sides of this equation by 60 to get:
r = -240p + p*c + 960 - 4c
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since c = -60p + 240, we substitute for c by replacing it with -60p + 240 in the equation to get:
r = -240p + p*(-60p+240) + 960 - 4(-60p+240)
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this becomes:
r = -240p - 60p^2 + 240p + 960 +240p - 960
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this reduces to:
r = -60p^2 + 240p
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our equation for revenue is r = -60p^2 + 240p
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to find the roots, we set it equal to 0.
-60p^2 + 240p = 0
multiply both sides of the equation by (-1) to get:
60p^2 - 240p = 0
divide both sides of the equation by 60 to get:
p^2 - 4p = 0
factor out the p to get:
p*(p-4) = 0
the roots are:
p = 0
or
p = 4
these would be the points where revenue would = 0 which means that the graph crosses the x axis.
from the equation you see that when p = 0 r = 0.
from the equation you see that when p = 4, r = 0
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since this is a quadratic equation, we can find the maximum/minimnum point by taking -b/2a.
standard form of quadratic equation is ax^2 + bx + c = 0
our equation is p^2 - 4p = 0 (the p stands for x).
this makes:
a = 1
b = -4
c = 0
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-b/2a = 4/2 = 2
our equation will be at a maximum or at a minimum when p = 2.
it turns out to be a maximum, so we have maximum revenue with p = 2.
when p = 2, c = 120.
a graph of the equation is shown below.
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the equation we are graphing is the final equation for revenue which was:
r = -60p^2 + 240p
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to graph the equation, we set
x = p = price of lemonade
and we set the y = c = quantity of cups of lemonade sold
values on the x-axis becomes the price of lemonade and values on the y-axis become the quantity of cups of lemonade sold.
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this problem was not tremendously difficult to solve, but it was tricky.
you had to keep very close attention to the numerous sign changes that occurred.
you needed to know the formula for a straight line.
you needed to know the formula for a quadratic equation.
you needed to know the formula for the maximum/minimum point of a quadratic equation.
you needed to know that you could get the formula of p from the formula for c, or that you could get the formula of c from the formula for p.
you needed to know that you could substitute c for p, or p for c, in the final equation to eliminate one of the unknowns.
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