Chapter 4.6, Problem 28PPE

To determine

Function of the cycle of a machine and domain and range of the function

Answer to Problem 28PPE

The function is : $y=\frac{1}{6}+0.25x$

• The domain of function is: $0\le x\le 23.3$

The range of function is: $0\le y\le 6$

• The function is a linear function.

Explanation of Solution

Given data:

Time taken for warm up - $10\min \text{ or }\frac{\text{1}}{\text{6}}hours$

Time taken for one cycle- 15 min or 0.25 hours

Total time of running in one day-6 hours

Calculation:

As the total time for running a machine in one day is 6 hours and its warm up time is 1/6 hours and for one cycle it takes 0.25 hours than the equation is:

  $y=\frac{1}{6}+0.25x$

Where y = total time in one day

X = number of cycle.

So,

  $0\le x\text{ (less than zero cycle can`t be performed)}$

As the one cycle of machine is completed in 15 min

It takes 10 min to warm up

Machine work in one day is 360 min

So,

1. $\begin{array}{l}\frac{\text{ total time of working -time of warm up}}{\text{time taken for completing one cycle}}\\ \frac{360-10}{15}\\ \frac{350}{15}\\ 23.3\\ so\\ x<23.3\end{array}$

The domain of function is:

  $0\le x\le 23.3$

  $\begin{array}{l}For,\\ 0\le x\\ \text{if there will be no waork done}\\ \text{than machine will give you zero output }\\ \text{So,}\\ \text{y}\le \text{0}\end{array}$

  $\begin{array}{l}\text{Maximum range of machine is when }\\ \text{x}\le \text{23}\text{.3}\\ \text{y}\le \frac{1}{6}+23.3(0.25)\\ \text{y}\le 6\end{array}$

The range of function is:

  $0\le y\le 6$

2. The function is a linear function.

  

Conclusion:

The function is : $y=\frac{1}{6}+0.25x$

• The domain of function is: $0\le x\le 23.3$

The range of function is: $0\le y\le 6$

• The function is a linear function.