Speed, distance and time formulas
$\text{average speed}=\frac{\text{total distance travelled}}{\text{total time taken}}$average speed=total distance travelledtotal time taken
$S=\frac{D}{T}$S=DT
| Unknown time | Unknown distance |
|---|---|
| $T=\frac{D}{S}$T=DS | $D=S\times T$D=S×T |
Map scales
To create maps, building plans, and other technical drawings, the features being represented must be scaled down to fit on the piece of paper, and we express this scaling factor with a ratio. For example, if a small city is $100000$100000 times larger than a piece of paper, scaling its features down onto a map drawn on that paper would have the scaling ratio of $1:100000$1:100000, meaning $1$1 cm measured on the map represents $100000$100000 cm (or $1$1km) in real life.
Another way to represent the distances on a map or building plan is to use a scale bar. This small bar on the drawing shows the corresponding distance in real life. On a map, a scale bar might measure $10$10 cm long, but if it is labelled as $20$20 km we know that if two features are $10$10 cm apart on the map then they are $20$20 km apart in real life.
Worked example
Example 1
Felicia is traveling directly from Cleveland to Pittsburgh. Approximately how far does she need to travel, and in what direction?
Think: By looking at the compass rose in the top right corner of the map, we can see that Felicia needs to travel in the South East direction. To find the distance we need to measure the distance on the map between Cleveland and Pittsburgh, and measure the length of the scale bar in the bottom left corner. Then we will divide the distance by the length of the bar and multiply by $100$100 to get the number of kilometres.
Do:
| Distance between Cleveland and Pittsburgh on map | $=$= | $3.4$3.4 cm |
| Length of scale | $=$= | $1.9$1.9 cm |
| Distance between Cleveland and Pittsburgh | $\approx$≈ | $\frac{3.4}{1.9}\times100$3.41.9×100 km |
| $\approx$≈ | $180$180 km |
So Felicia needs to travel approximately $180$180 km in the South East direction.
Note: Measurements in centimetres may vary depending on your display.
Practice question
Question 1
Carl wanted to fly from Perth to Adelaide with an overnight stop in Sydney.
Carl's first plane took $4$4 hours of flight time. If the airplane was travelling at an average speed of $825$825 km/h, what is the distance between Perth and Sydney as it appears on the map, measured in scaled units on the map?
Give your answer to one decimal place.
Carl's second trip took $2.2$2.2 hours. If the distance between Sydney and Adelaide appears to be $2.2$2.2 units on the map, what was the average speed of the airplane? Give your answer to the nearest integer.
Carl's overnight stay at a hotel in Sydney costs $$120$120. If the flight company charges $11$11 cents/km for the flight, what was the total cost of the trip?
Give your answer to the nearest cent.