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Question

A classroom teacher has to purchase new markers because public education is radically underfunded. In an effort to predict how many markers he should purchase, this teacher builds a function for predicting when he will run out. He has found, over the year, that he uses up to 2 markers each week. With 180 days of school at the beginning of the school year, he hopes to purchase markers once in July and have enough to make it to the end of the year.

Write a function, in function notation, that describes this situation and use it to predict how many markers the teacher needs to purchase in July to make it through the school year.

Answer 1

In this question we are asked to create a function that will help the teacher predict the amount of markers needed to make it through the school year.

One important idea is that a function has an input and an output. (x and f(x))

We know that our output should represent the # of markers needed to be purchased. With this in mind, what should be our input value? What affects the amount of markers that the teacher needs? The input is the amount of time or length of the school year. This makes sense because if the school year was shorter, you would need fewer markers, etc. Because the the length of the school year is given as 180 days, our input will be in days.

Equation setup:

Let x = # of days in school year.

Let f(x) = # of markers needed

We know that we expect to use 2 markers per week. Because this is given in terms of weeks and our x variable is in terms of days, we need to divide by seven (there are 7 days in a week). We also know that in each of these 7-day period we need 2 markers, thus we end up with the following model:

f(x) = 2x / 7

To check that the model makes sense, plug in the most basic case. If school was only one week (7 days) long, we know we would only need 2 markers. So, plug in 7 for x (7 days) into the function and we should get an output of 2 markers.

f(7) = 2(7) / 7

=14 / 7

=2

To predict the amount of markers the teacher needs for 180 days, plug in 180 for x:

f(180) = 2(180) / 7

= 360 / 7

= 51.42857143

Here we would need one extra marker to since the answer is >51, so based on the model the teacher should purchase 52 markers for a 180-day school year.

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