This graph shows how the height of a lift changed over time. Estimate the speed of the lift after 24 seconds. Give your answer in meters per second. Show your working.
Before starting any graph question, it is important that we understand what each little square on the axes represents.
On the 𝑦-axis, we have the height measured in meters. One square is equal to two meters. On the 𝑥-axis, we have the time in seconds. Five squares here are equal to 20 seconds.
Dividing both of these by five tells us that one square is equal to four seconds as 20 divided by five is equal to four.
We were asked to estimate the speed of the lift after 24 seconds. Well, speed equals distance divided by time. Our units for distance were meters and our units for time were seconds. Therefore, our units for speed will be meters per second.
In order to estimate the speed after 24 seconds, we need to draw a tangent at 24 seconds. We then select two points on this tangent and calculate the speed by dividing the change in distance by the change in time.
24 seconds is one square to the right of 20 seconds as we said that one square was equal to four seconds. This means that the point at which we need to draw our tangent has coordinates 24, 18. The time is equal to 24 seconds and the height is equal to 18 meters.
A tangent is a straight line that touches the graph at a particular point. We now need to pick a second point or coordinate on this tangent line. For this example, we will pick zero, eight. However, you could choose any other points on the line.
Note at this point that not everyone’s tangent line will look the same. There will be a small margin for error at this point. We now need to estimate the speed by calculating the change in distance and the change in time.
This is also equal to the gradient, which you might have seen written as rise over run. We can draw our right-angled triangle joining these two points and then calculate the rise, the run and divide the two answers.
The change in distance or the rise is calculated by subtracting eight from 18 as the two 𝑦-coordinates are eight and 18. In the same way, the change in time or the run can be calculated by subtracting zero from 24 as these were the two 𝑥-coordinates.
18 minus eight is equal to 10 and 24 minus zero equals 24. Therefore, the speed is equal to 10 over 24 or ten twenty-fourths. We can simplify this fraction by dividing the numerator and denominator by two.
10 divided by two is equal to five and 24 divided by two is equal to 12. We could also turn our answer into a decimal by dividing five by 12. This gives us 0.41666 and so on. Rounding this answer to three decimal places is 0.417.
This means that an estimate for the speed of the lift after 24 seconds is 0.417 meters per second.
Once again, it’s important to note that your answer might be slightly different. But you’ll get the mark if you find the gradient of your tangent.