Calculate a rate from a straight-line graph

Point A

\(\displaystyle (1, 5)\)

Point B

\(\displaystyle (4, 14)\)

A hiker walks along a straight path away from a fixed marker at a steady pace. The graph shows distance from the marker against elapsed time. What is the walking rate?

The graph records distance travelled since an earlier starting point against time since this recording began. Move B and compare the ratio of changes with the ratio of B’s coordinates

As point B moved along the line, for every position of the slider, what happened to the readouts?

Rate from a graph

The rate on a straight-line graph is the vertical change divided by the horizontal change between any two points on the line. It has vertical units per horizontal unit, and it is the same whichever two points are used

1. 1Select two points Use two clear points on the line and record their horizontal and vertical coordinates \(\displaystyle A(1, 5),\quad B(4, 14)\)
2. 2Find the vertical change Subtract the earlier vertical value from the later vertical value, keeping its sign \(\displaystyle \Delta y=14-5=9\ \mathrm{km}\)
3. 3Find the horizontal change Subtract the earlier horizontal value from the later horizontal value \(\displaystyle \Delta x=4-1=3\ \mathrm{h}\)
4. 4State the rate with units Divide vertical change by horizontal change and express the result as vertical units per horizontal unit \(\displaystyle r=\frac{\Delta y}{\Delta x}=\frac{9}{3}=3\ \mathrm{km/h}\)

Point P

\(\displaystyle (2, 900)\)

Point Q

\(\displaystyle (8, 300)\)

A tank drains steadily. The graph shows volume remaining against elapsed time, in minutes and litres. Find the signed rate of change of volume

Find both signed changes from P to Q, keep the sign, then divide the water change by the time change

Point P

\(\displaystyle (2, 900)\)

Point Q

\(\displaystyle (8, 300)\)

Rate of change

\(\displaystyle -100\ \mathrm{L/min}\)

If the tank keeps draining at the same rate, at what time is the tank empty?

Use the rate in reverse: find how much water is left after point Q, then find how long that amount takes to drain

Rise

\(\displaystyle 9\ \mathrm{km}\)

Run

\(\displaystyle 3\ \mathrm{h}\)

Using the units in the wrong order. One of these lines is right. Which one?

A tank contains 900 L. Its constant outflow doubles. What happens to the time needed to empty it?

At A the tank contains 900 L. Change the outflow and observe the segment, signed rate and time remaining. Compare 50 L/min with 100 L/min.

What happens when the outflow doubles from 50 to 100 L/min?

The volume in a tank changes at −100 L/min. What is its positive outflow rate?

Volume remaining

\(\displaystyle 300\ \mathrm L\)

Outflow

\(\displaystyle 100\ \mathrm{L/min}\)

At time 8 min there are 300 L left, draining at 100 L/min. How many more minutes are needed to empty the tank?

A constant-drain model predicts −200 L after the tank has emptied. How should this be interpreted?

Given rate

\(\displaystyle 90\ \mathrm{L/min}\)

A pump supplies 90 L in each minute. Express its rate in litres per second

Pump A supplies 1.5 L/s. Pump B supplies 80 L/min. Which has the greater flow?

First reading

\(\displaystyle (2,6)\)

Second reading

\(\displaystyle (6,16)\)

A process transfers 6 millicoulombs of electric charge after 2 items and 16 millicoulombs after 6 items. Find the additional charge per item

A horizontal volume-time segment stays at 240 L for four minutes. What is its rate of change?

Use the model within its limits

A straight segment has a constant rate. The rate connects changes; the initial value tells you where to start

1. 1Forward application Multiply the rate by the horizontal change, then add the vertical change to the starting value \(\displaystyle \Delta y=r\Delta x\)
2. 2Reverse application For a nonzero rate, divide the vertical change by the rate to obtain the horizontal change \(\displaystyle \Delta x=\Delta y\div r\)
3. 3Read the requested time A duration is a change in time. Add it to the starting clock reading only when an absolute time is requested
4. 4Curved graphs The ratio of endpoint changes gives an average rate over the interval. It does not establish a constant rate within that interval