Question

In a factory, a certain machine needs 10 min to warm up. it takes 15 min for the machine to run a cycle. the machine can operate for as long as 6 h per day including warm up time. Draw a graph showing the total time the machine operates during 1 day as a function of the number of cycles it runs.

-What domain and range are reasonable?
-Is the function a linear function?

Answer 1

`y =(1)/(6) + 0.25x`

`0 \leq x \leq23.33`

`0 \leq y \leq 6`

Explanation:

The machine always takes 10 minutes (or 1/6 hours) to warm up and performs each cycle in 15 minutes, or 0.25 hours.

The maximum amount of daily hours that the machine can work is 6 hours.

Therefore we can represent this situation by means of an equation of the form:

`y =(1)/(6) + 0.25x`

Where

y = daily amount of hours the machine works.

x = number of daily cycles performed.

This equation is a linear function. Because it has the form y = ax where a is a real number

The graph of this equation is shown in the attached image, where the gray shaded area represents the domain of the function.

The domain of the function is:

`x\geq 0` (because you can not perform less than zero cycles a day)

`x \leq23.33`. Because the machine can not exceed 6 hours per day, then it can not do more than 23.33 cycles.

So:

`0 \leq x \leq23.33`

The range of the function is:

`y \geq 0` (because you can not work less than 0 hours a day)

`y \leq 6` (because the machine can not work more than 6 hours a day)

So

`0 \leq y \leq 6`