When selling lemonade for 50 cents per cup, Diana can sell 40 cups per day. She observes that the number of cups sold per day decreases by 2 for every 5-cent increase in the price of a cup of lemonade.
Determine a formula for the number cups C of lemonade sold each day as a function of the price p in cents of a cup of lemonade if all price increases occur only in multiples of 5-cent increments.
c = cups/day
p = pennies/cup
dc/dp = -2/5
so
c = -(2/5) p + constant
when p = 50, c = 40
so
40 = -2*50/5 + constant
constant = 60
and
c = -(2/5) p + 60
now you can make that p = 5n if you choose where n is an integer
number of nickels
c = -2 n + 60
To determine a formula for the number of cups sold each day as a function of the price of a cup of lemonade, let's break down the given information step by step:
1. We are given that Diana can sell 40 cups of lemonade per day when the price is 50 cents per cup. This gives us an initial data point: C = 40 when p = 50.
2. It is stated that the number of cups sold per day decreases by 2 for every 5-cent increase in the price of a cup of lemonade. This means that for each 5-cent increase, the number of cups sold decreases by 2. Therefore, we can establish a pattern where the number of cups sold decreases by 2 for every 5-cent increment.
To determine the formula, we need to determine the relationship between the number of cups sold and the price of a cup of lemonade. Let's express this relationship as follows:
Cups sold = f(price) = f(p)
To start, we know that when the price is 50 cents (p = 50), Diana sells 40 cups. So we have the initial condition:
f(50) = 40
Next, we need to incorporate the information that the number of cups sold decreases by 2 for every 5-cent increase in price. We can express this relationship as a function:
f(p) = 40 - [(p - 50) / 5] * 2
Let's break down this formula:
- (p - 50) represents the difference between the actual price and the initial price of 50 cents.
- Dividing this difference by 5 gives us the number of 5-cent increments.
- Multiplying this by 2 represents the decrease in the number of cups sold for each 5-cent increment.
- Finally, subtracting this decrease from the initial number of cups sold (40) gives us the updated number of cups sold based on the price increment.
Therefore, the formula for the number of cups sold (C) as a function of the price (p) is:
C = 40 - [(p - 50) / 5] * 2