6. Suppose you have a lemonade stand, and when you charge $2 per cup of lemonade you sell 100 cups. But when you raise your price to $4 you only sell 50 cups. Write an equation for the number of cups you sell as a function of the price you charge. Denote "C" for number of cups, and "P" for the price you charge. Assume the function is linear. Show your work.
Well, my dear friend, let me entertain you with my mathematical comedic skills!
First, let's establish the relationship between the number of cups sold (C) and the price charged (P). We are told that when the price is $2, you sell 100 cups, and when it's $4, you sell 50 cups. So, we have two sets of data: (P = 2, C = 100) and (P = 4, C = 50).
Now, let the slope of our linear function be represented by "m" and the y-intercept be denoted by "b". To find the equation, we'll substitute the given points into the slope-intercept form equation, y = mx + b.
First, let's calculate the slope (m):
m = (C2 - C1) / (P2 - P1)
= (50 - 100) / (4 - 2)
= -50 / 2
= -25
Great! Now we know the slope is -25. Let's find the y-intercept (b) by substituting one of the given points:
100 = -25 * 2 + b
Solving for b, we get:
100 = -50 + b
b = 150
Hilarious, isn't it? Now we have our slope (m = -25) and y-intercept (b = 150). So, the equation for the number of cups sold (C) as a function of the price charged (P) is:
C = -25P + 150
Now, if someone asks you how many cups you'll sell if you charge $6 per cup, you can plug it into the equation and find out!