Question

Write a linear model for the amount of boxes, b, as a function of the number of hours since they opened, h. Use your model to predict the number of boxes in stock at the end of an 8 hour shift.

Answer 1

Question:

A company produces boxes of DVDs at a rate of 52 boxes per hour they begin to produce boxes when they first opened for the day and after 3 hours have 400 boxes in stock.

- How many boxes were in stock when they opened?

- Write a linear model for the amount of boxes, b, as a function of the number of hours since they opened, h.

- Use your model to predict the number of boxes in stock at the end of an 8 hour shift.

Answer:

1.
`Initial = 244`

2.
`b = 244 + 52h`

3. 660 boxes

Explanation:

Given

`Rate = 52\ per\ hour`

`Total\ boxes\ in\ 3\ hours = 400`

Solving (a): Initial Number of boxes

First, we calculate the number of boxes made in 3 hours

`Boxes = Rate * Hours`

`Boxes = 52 * 3`

`Boxes = 156`

If they had 400 boxes at the 3rd hour.

Then, the number of boxes when they opened is:

`Initial = 400 - 156`

`Initial = 244`

Solving (b): Linear function

`b = boxes`

`h = hours`

Since, we have the rate and the initial number of boxes.

The linear function is:

`Boxes = Initial + Rate * Hours`

i.e.

`b = 244 + 52 * h`

`b = 244 + 52h`

Solving (c): Boxes at the end of the 8th hour

We have:

`b = 244 + 52h`

In this case:

`h = 8`

Substitute 8 for h

`b = 244 + 52 * 8`

`b = 244 + 416`

`b = 660`

Hence, there are 660 boxes at the end of the 8th hour