Video: AQA GCSE Mathematics Higher Tier Pack 4 • Paper 3 • Question 23

Two mathematically similar cereal boxes are made of card. The volume of the smaller box is 2250 centimeters cubed, and the volume of the bigger box is 72.8 percent larger than the smaller box. The height of the smaller box is 30 centimeters. Calculate the height of the bigger box.

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Video Transcript

Two mathematically similar cereal boxes are made of card. The volume of the smaller box is 2250 centimeters cubed, and the volume of the bigger box is 72.8 percent larger than the smaller box. The height of the smaller box is 30 centimeters. Calculate the height of the bigger box.

If two three-dimensional objects are mathematically similar, then this just means that one is an enlargement of the other. All of their pairs of corresponding side lengths will be in the same ratio. In this question, we’ve been told the height of the smaller box. It’s 30 centimeters. And we’ve been asked to calculate the height of the bigger box, which means we need to know the ratio between the lengths of the two boxes. We can refer to this as the length scale factor or LSF.

The other information that we’re given is that the volume of the smaller box is 2250 centimeters cubed and the volume of the bigger box is 72.8 percent larger. Let’s think about how we can use this information to help.

Now you may think that what we need to do is calculate the volume of the larger box, but actually we don’t need to. What we’re going to do is look at the ratio between the volumes of the two boxes. The volume of the larger box is 72.8 percent more than the volume of the smaller box, which means it’s 172.8 percent of the volume of the smaller box. If we were going to work this out, we’d have to find 172.8 percent of 2250.

One way to do this calculation would be to use a decimal multiplier. If we want to find 172.8 percent of a number, we can multiply that number by the decimal 1.728. This is because the word “of” can be replaced with “multiply” in a calculation involving percentages, and 172.8 percent literally means 172.8 out of 100, which is equivalent to 1.728. This means that, to scale up the volume of the smaller box to give the volume of the bigger box, we multiply it by 1.728, which we can refer to as a volume scale factor or a VSF for short.

So now we know the volume scale factor between these two boxes. But the question is, is the length scale factor the same? Well, actually it isn’t. And to see why not, let’s picture a simple example with a cube. In one case, we have a cube with a side length of one and in the second a cube with a side length of three. The length scale factor between these two cubes will be three, as the side lengths of the larger cube are three times bigger than those on the smaller cube.

If we calculated the volumes of the two cubes by multiplying their three dimensions together, then for the smaller cube, we’d have one centimeter cubed. That’s one times one times one. And for the larger cube, we’d have 27 centimetres cubed. That’s three times three times three. The volume scale factor between these two cubes then is not three but 27, as the larger cube’s volume is 27 times bigger than the smaller. There is a connection between 27 and three however. 27 is equal to three cubed. This is illustrative of a general rule which we can apply, which is that if the length scale factor between two similar solids is some number 𝑘, then the volume scale factor is 𝑘 cubed.

Incidentally, if we were asked something about the areas of two similar solids, then the area scale factor would be 𝑘 squared. Working back the other way then, if the volume scale factor, 𝑘 cubed, is 1.728, then the length scale factor, 𝑘, is the cubed root of this value. We can use a calculator to work this out. And it tells us that the length scale factor is equal to 1.2. This tells us then that the lengths on the bigger box are each 1.2 times larger than the corresponding lengths on the smaller box.

So to find the height of the larger box, we can take the height of the smaller box and multiply it by 1.2. This is equal to 36, so we found that the height of the bigger box is 36 centimetres.

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