Video: Pack 5 • Paper 2 • Question 10

Pack 5 • Paper 2 • Question 10

08:12

Video Transcript

The diagram shows a boat modelled as a prism of height 50 centimetres. The cross section of the prism is made of two trapeziums and one triangle. Lila is in the boat on the sea. There is a storm with rain and high waves. Water is coming into the boat. After 10 minutes, there are 25 litres of water in the boat. Assume that the water continues to come into the boat at this rate. Work out how many more minutes it takes for the boat to be 15 percent full of water. And we’ve got a reminder that one litre is equal to 1000 cubic centimetres.

So we’ve got a 3D sketch of the boat, showing that it’s 50 centimetres deep. And then, we’ve got an overhead view of the boat — a plan view of the boat — showing the shapes. We’ve got one trapezium, which has two parallel sides, which are 96 centimetres and 125 centimetres and the distance between them is 36 centimetres.

Our next trapezium has parallel sides of 125 centimetres and 108 centimetres and the distance between those is 40 centimetres. And then, we have a triangle with a base of 108 centimetres and a perpendicular height of 132 centimetres.

Now, some of the key information in the question is that after 10 minutes, there are 25 litres of water in the boat. So that’s the rate at which the water is pouring into the boat. And we’re going to assume that the water pours into the boat at that constant rate. So we’ve got to work out how many more minutes it takes for the boat to be 15 percent full of water.

So to answer our question, now, first, we need to find the volume of the whole boat. Then, we need to work out what 15 percent of that volume is. Then, we need to work out how long would it take to fill up that 15 percent. Then, we need to take off the 10 minutes that have already passed in putting the first 25 litres of the water to work out how many more minutes it takes to fill up to 15 percent.

Okay, I’m gonna summarize the key facts and then clear some space so we can do our calculations. So step one, to find the volume: the volume of a prism remember is the cross-sectional area times the depth of the prism. So first of all, we need to work out the area of the base of the boat. And I’m gonna split this into three. So this is trapezium number one, this is trapezium number two, and this is the triangle, which we’ll just call 𝑇 three.

So the total volume of the boat is gonna be 𝑇 one plus 𝑇 two plus 𝑇 three, which is the cross-sectional area times the depth. First, let’s recall that the area of a trapezium is the mean length of the parallel sides. So that’s 𝑎 plus 𝑏 all divided by two in this case times the height of the trapezium, the distance between those two parallel sides.

Now, all of our units are in centimetres. So we can just plug those straight into that formula: the area of 𝑇 one is gonna be a half of 96 plus 125 and then we’ll multiply that by 36. When we plug that into our calculator, we get 3978 square centimetres. So remember that was a half of this plus this all times this.

And for the area of 𝑇 two, we’re gonna do a half of this plus this all times this. And when we do that, we get an answer of 4660 square centimetres. Now, before working out 𝑇 three, let’s recall the formula for the area of a triangle: it’s a half times the triangle’s base times its perpendicular height.

Well, this particular triangle is kind of upside down. Here’s the base and here’s the perpendicular height. So it’s gonna be a half times 108 times 132. And when I plug those numbers into my calculator, I get an answer of 7128 square centimetres.

So to work out the volume, I need to add 𝑇 one, 𝑇 two, and 𝑇 three. So that’s 3978, 4660, 7128 and multiply that answer by the depth of the boat which is 50 centimetres. Now, if you added 𝑇 one, 𝑇 two, and 𝑇 three, you should have got 15766. And that makes the total volume 788300 cubic centimetres.

Now, we’ve got to work out 15 percent of that volume. And to work out 15 percent of something, it’s just fifteen hundredths of that value. And fifteen hundredths of 788300 is 118245. So the volume of water that we’re trying to achieve is exactly that.

Now, that’s all well and good. But look the rate that we’ve been given for the water pouring in is in litres per minute, not in cubic centimetres per minute. So we need to convert that volume into litres.

Now, we know that 1000 cubic centimetres gives us one litre. So how many litres will 118245 cubic centimetres give us? In other words, what do I need to multiply a 1000 by to get 118245? Well, a 1000 times something is equal to 118245. If I divide both sides of that equation by 1000, then the thousands will cancel out on the left-hand side, just leaving us with our missing number and. And we’re left with 118.245 on the right-hand side. So that’s our missing number.

Now, if our volume in cubic centimetres is 118.245 times bigger than a 1000 cubic centimetres, then it’s the same number of times bigger than one litre. And one times 118.245 is obviously 118.245. So that’s the number of litres.

Now, we need to know how long it’s gonna take us to pour in 118.245 litres of water. So if 25 litres takes 10 minutes, how many minutes does it take for 118.245 litres? How do we turn 25 into 118.245? Well, there’s a surefire, two-step method. First, divide 25 by itself, by 25. That’s gonna give us one. Then, we got to turn one into 118.245. Well, we can multiply it by 118.245 to do that.

So if that’s the path we’ve done down the left-hand side with the number of litres, we’ve got the same proportion of things happening on the right-hand side to the minutes. So 10 divided by 25 is 10 over 25. And we’ll multiply that number by 118.245. And our calculator tells us that that is 47.298 minutes. Then, if we take off the 10 minutes that have already passed, that leaves us with 37.298 minutes more to fill 15 percent of the boat with water. Now, we may choose to round that down to exactly 37 minutes. We wouldn’t normally say it’s 37.298 minutes, but that’s the answer.

Now, the assumption about the rate at which the water is coming into the boat could be wrong. So for part b of this question, we got to explain how this could affect your answer to part a. Well, if our assumption was wrong, there are two possible cases: the rate could be higher — that’s more litres of water per minute flowing into the boat — or it could be lower — that’s fewer litres of water per minute flowing into the boat.

Now, if the rate was higher, more water is coming in in the same amount of time. It’s gonna take less time to fill 15 percent of the boat with water. And if the water rate was lower, so fewer litres of water were coming in per minute, then obviously it’s gonna take longer more time to fill 15 percent of the boat with water. And we need to mention both of those two cases.

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