Finding Linear Equations in Context
Now that we know how
- to graph linear relationships
- to find the equations of linear functions
- to use algebra and graphs to extract information
- to find intercepts and constant values, and
- that the gradient of a linear function represents constant change.
we can put this to use to solve a range of real life applications.
It's all the same mathematics, but this time you will have a context to apply it to.
Some examples will be the best way to show you how the mathematics we have learnt can be applied to everyday situations.
Examples
Question 1
A carpenter charges a callout fee of $\$150$$150 plus $\$45$$45 per hour.
Write an equation to represent the total amount charged, $y$y, by the carpenter as a function of the number of hours worked, $x$x.
What is the gradient of the function?
What does this gradient represent?
The total amount charged increases by $\$45$$45 for each additional hour of work.
AThe minimum amount charged by the carpenter.
BThe total amount charged increases by $\$1$$1 for each additional $45$45 hours of work.
CThe total amount charged for $0$0 hours of work.
DWhat is the value of the $y$y-intercept?
What does this $y$y-intercept represent?
Select all that apply.
The total amount charged increases by $\$150$$150 for each additional hour of work.
AThe maximum amount charged by the carpenter.
BThe callout fee.
CThe minimum amount charged by the carpenter.
DFind the total amount charged by the carpenter for $6$6 hours of work.
Question 2
The table shows the linear relationship between the length of a mobile phone call and the cost of the call.
| Length of call (minutes) | $1$1 | $2$2 | $3$3 |
|---|---|---|---|
| Cost (dollars) | $7.6$7.6 | $14.4$14.4 | $21.2$21.2 |
Write an equation to represent the cost of a call, $y$y, as a function of the length of the call, $x$x.
What is the slope of the function?
What does the slope tell you?
The connection fee
AThe cost of each additional minute
BThe cost of the phone
CThe cost of a $1$1-minute call
DWhat is the $y$y-intercept?
What does this $y$y-intercept tell you?
The cost of each additional minute
AThe cost of the phone
BThe cost of a $1$1-minute call
CThe connection fee
DFind the cost of a $6$6-minute call.
Question 3
The graph shows the amount of water remaining in a bucket that was initially full before a hole was made in its side.
A line graph has its $x$x-axis labeled "time (minutes)" and $y$y-axis labeled "quantity (litres)". The $x$x-axis ranges from $0$0 to $40$40, marked at major intervals of $4$4 minutes and minor intervals of $2$2 minutes. The $y$y-axis ranges from $0$0 to $40$40, marked at major intervals of $4$4 litres and minor intervals of $2$2 litres. A line is plotted on the line graph. The line starts at $\left(0,30\right)$(0,30) and moves downward from left to right. The line also passes through the point $\left(12,24\right)$(12,24) and extends beyond the visible part of the graph. The points are not marked, and their coordinates are not explicitly labeled or given.
What is the gradient of the function?
What is the $y$y-value of the $y$y-intercept?
Write an equation to represent the amount of water remaining in the bucket, $y$y, as a function of time, $x$x.
What does the slope tell you?
The amount of water remaining in the bucket after $2$2 minutes.
AThe amount of water that is flowing out of the hole every minute.
BThe time it takes for the bucket to be completely empty.
CThe time it takes the amount of water remaining in the bucket to drop by one litre.
DWhat does the $y$y-intercept tell you?
The capacity of the bucket.
AThe amount of water remaining in the bucket after $30$30 minutes.
BThe amount of water remaining in the bucket when it is empty.
CThe size of the hole.
DFind the amount of water remaining in the bucket after $54$54 minutes.