In order to gain popularity amongst high school students the local coffee shop is delivering coffee to sell at lunch. Below is a function that represents the prices of coffee where t is the days since January 10th.
Piecewise function
c(t) =
2; 1<= t<=3
2 +t; 3<= t<=4
5; t =>4
1 What are the various prices of coffee?
= 3, 2 + t, 5
2. Which domain does the 4th day fit in?
3. How much is coffee on the 4th day?
2 + 4 = 6
4. Which domain does the 5th day fit in? How much would the coffee be?
t=>4
=$5
5. What does c (3) mean?
c(3) = $3
6. What days does the coffee cost $2?
Days 3
7. What would be the cheapest day(s) to buy a coffee?
8. What is the most expensive day(s) to buy a coffee?
9. What is the value of c(3) - c(1)
To find the answers to the questions, we need to evaluate the piecewise function for the given values of t.
1. The various prices of coffee are: $3 for t between 1 and 3, $2 + t for t between 3 and 4, and $5 for t greater than or equal to 4.
2. The 4th day fits in the domain where t is less than or equal to 4, so it falls into the second part of the piecewise function.
3. To find the price of coffee on the 4th day, substitute t = 4 into the second part of the function: 2 + 4 = $6.
4. The 5th day also falls into the third part of the function because t is greater than or equal to 4. Therefore, the price of coffee on the 5th day would be $5.
5. c(3) means evaluating the function at t = 3. Substituting t = 3 into the first part of the function, we get c(3) = $3.
6. To find the days when the coffee costs $2, we need to look at the second part of the function. It is between t = 3 and t = 4, so the coffee costs $2 on day 3.
7. To determine the cheapest day(s) to buy a coffee, we need to find the minimum price among all the given prices. In this case, the coffee costs $2 on day 3, which is the cheapest price.
8. To find the most expensive day(s) to buy a coffee, we need to find the maximum price among all the given prices. The coffee costs $5 for t greater than or equal to 4, so the most expensive day(s) to buy a coffee would be on or after the 4th day.
9. To calculate c(3) - c(1), we substitute t = 3 and t = 1 into the function separately. c(3) = $3 and c(1) = $2. Therefore, c(3) - c(1) = $3 - $2 = $1.
The value of c(3) - c(1) would be:
c(3) = $3
c(1) = $2
Therefore, c(3) - c(1) = $3 - $2 = $1.