Zoe has a water gun with a full tank of water. Below is some data on the flow rate of one shot from the water gun. Assume all shots of the water gun follow the same profile. Zoe completes 24 full shots of the water gun before the tank is completely empty. Zoe is carrying a bottle with exactly 340 millilitres of water in it. Is the water in this bottle enough to completely fill up the tank of the water gun? You must show your working.
Before we start this question, we need to understand what the graph is showing. On the 𝑦-axis, we have the flow rate in millilitres per second. On the 𝑥-axis, we have the time in seconds. We need to calculate the volume in millilitres.
This graph is very similar to a velocity-time graph, where the velocity would be on the 𝑦-axis measured in metres per second and the time would be on the 𝑥-axis measured in seconds. In order to work out the distance on a velocity-time graph, we work out the area under the graph. This is because velocity or speed multiplied by time is equal to distance.
Multiplying the units metres per second and seconds gives us an answer of metres. We can work out the volume of water in the same way in this question. We can multiply the flow rate by the time as millilitres per second multiplied by seconds gives us a unit of millilitres.
We can, therefore, say that the area under the line of the graph is equal to the volume in millilitres of one shot.
The graph is in the shape of a trapezium. And the area of a trapezium can be calculated by multiplying a half by 𝑎 plus 𝑏 by the height. 𝑎 and 𝑏 are the parallel sides of the trapezium and ℎ is the perpendicular height between them.
In our graph, 𝑎 is equal to one. It is the time between 0.5 seconds and 1.5 seconds. 𝑏 is equal to 2.5, the time between zero and 2.5 seconds. ℎ is equal to eight as it is the flow rate between zero and eight.
We can substitute these values into the formula for the area of a trapezium to calculate the volume in one shot. The volume is equal to a half multiplied by one plus 2.5 multiplied by eight.
A half multiplied by eight is equal to four and one plus 2.5 equals 3.5. Therefore, we need to multiply four by 3.5. This is equal to 14. Therefore, the volume in one shot of the water gun is 14 millilitres.
Zoe completed 24 full shots before the tank was empty. Therefore, we need to calculate the volume in 24 shots. We need to multiply 14 millilitres by 24. This is equal to 336. Therefore, the total volume of water in 24 shots is 336 millilitres.
Zoe’s water bottle had 340 millilitres in it. 340 is greater than 336. This means that yes, Zoe does have enough water to completely fill up the tank. She needed 336 millilitres to fill the tank and the bottle of water contained 340 millilitres.