How to solve a piecewise word problem

Question

Jonas owns a popular coffee shop called Java the Hut. The shop hours are 7 am to 8 pm and Saturdays are always the stores busiest days. Jonas was working hard to sell as much coffee as possible, so He opened up at exactly 7 and sold 80 cups of coffee in a 2 hour period, then the sales slowed down and he sold 46 cups of coffee in a 3 hour period. Suddenly, a large wave of people came in, and he sold 34 cups of coffee in an hour. The large wave trickled out and work slowed down, so no one came in for an hour. Then a few customers began to trickle in and out, and he sold 23 cups of coffee in a 2 hour period. Then no one came in for an hour, and he slipped on the wet floor, spilling 12 cups worth of coffee beans. People began to come back in after that, and he sold 35 cups of coffee in 2 hours. Then the rush died down and he sold 17 cups of coffee in an hour.

 

     I was told to make a problem but was unable to convert it into a piecewise function

 

Answer 1

The problem you have described is not necessarily a problem about creating a piecewise function, but it can written down as one. To do so we need to:

1. assume that Jonas sells the cups of coffee at a constant rate for each of the listed time periods,
2. define the base variable (current time)
3. define the changed value (this is tricky, but we can either choose the amount of beans used/lost or the amount of coffee cups sold)

Let's call the base variable t, and the changed value we will define through the function f(t). To utilize all of the listed information, let the changed value be the amount of beans used/lost. Then we need to carefully follow the described situation to define each piece of the piecewise function:

1. 0 ≤ t < 7 - Jonas used no beans. f(t) = 0, 0 ≤ t < 7
2. 7 ≤ t < 9 - Jonas used 80 cups worth of beans. Rate = 80/2 = 40 ⇒ f(t) = 40t + b, where b - y-intercept. We also have a point that belongs to the line describing this piece (7,0) ⇒ 0 = 40 * 7 + b ⇒ b = -280. f(t) = 40t - 280, 7 ≤ t < 9.
3. 9 ≤ t < 12 - 46 cups. Rate = 46/3 ⇒ f(t) = (46/3)t + b. (9, 80) ⇒ 80 = (46/3) * 9 + b ⇒ b = -58. f(t) = (46/3)t - 58, 9 ≤ t < 12

The rest can be done similarly.