Video: AQA GCSE Mathematics Foundation Tier Pack 2 • Paper 3 • Question 25

A cuboid has been removed from a larger cuboid, as shown in the diagram. Given that the cuboid removed is similar to the larger cuboid, show that the volume of the remaining solid is 3166 2/3 cm³.

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Video Transcript

A cuboid has been removed from a larger cuboid, as shown in the diagram. Given that the cuboid removed is similar to the larger cuboid, show that the volume of the remaining solid is 3166 and two-thirds centimeters cubed.

So this means the large cuboid has a small cuboid shape hole cut out of it. And the small cuboid shape is similar to the larger cuboid shape. That just means that all the sides are in the same proportions. Okay, let’s define some variables. We’re going to let 𝑉 large represent the volume of the large cuboid. And we calculate the volume of the large cuboid by multiplying the length by the width by the height. So that’s 15 centimeters times 15 centimeters times 20 centimeters which is 4500 cubic centimeters. And we’re going to let 𝑉 small represent the volume of the small cuboid. Well, we don’t know all the side lengths yet. So we can’t work that out just yet. And let’s let 𝑉 remain equal the volume of the remaining solid. And that’s just be volume of the large cuboid minus the volume of the small cuboid.

Now, let’s take a moment to identify the corresponding edges of our two similar cuboids. This bottom edge of the small cuboid corresponds to this bottom edge of the large cuboid. This bottom edge of the small cuboid corresponds to this bottom edge of the large cuboid. And the vertical height of the small cuboid corresponds to the vertical height of the large cuboid. So when we think about length, we know that 10 centimeters on the small cuboid corresponds to 15 centimeters on the large cuboid. And that means that lengths on the small cuboid are 10 15ths of the size of lengths on the large cuboid. And 10 and 15 are both divisible by five. So 10 divided by five is two and 15 divided by five is three.

With three-dimensional shapes, there are three corresponding scale factors that we have to think about. The length scale factor, we’ve just worked out is two-thirds. Lengths on the small cuboid are two-thirds of the lengths on the large cuboid. The area scale factor is the square of the length scale factor. So that’s two-thirds squared which is four-ninths. So corresponding areas on the small shape are only four-ninths as large as the areas on the large cuboid. So one side, for instance, is only four-ninths the area of the corresponding side on the large cuboid. And the volume scale factor is the cube of the length scale factor. And that is two-thirds cubed which is the same as two cubed over three cubed. Two cubed means two times two times two which is eight and three cubed means three times three times three which is 27. So that means that the volume of the small cuboid is only eight 27ths of the volume of the large cuboid.

So now, we can work out the value of 𝑉 small, the volume of the small cuboid, is eight 27ths times the volume of the large cuboid 4500 cubic centimeters. And my calculator tells me that’s 4000 over three cubic centimeters. Now, it might be tempting to represent that as a decimal. But there’s no point in doing that because the question wants us to use fractions. So we’re gonna leave this value in this format for now.

Well now that know the volume of the large cuboid and we know the volume of the small cuboid, we can work out the remaining volume when we take the small one away from the large one. So the remaining volume is 4500 cubic centimeters minus 4000 over three cubic centimeters. And my calculator tells me that’s 9500 over three cubic centimeters. Well, that’s not quite in the format that we’re looking for. Now, if you’re lucky, you’ll have a button on your calculator that converts top heavy fractions like we’ve got here, 9500 over three, into mixed numbers, 𝑎 plus 𝑏 over 𝑐. If you press that button, it tells you that it’s equivalent to 3166 and two-thirds. All we have to do is add the units, cubic centimeters. And we’ve shown that the volume of the remaining solid is 3166 and two-thirds cubic centimeters.

The first step then was to work out the volume of the large cuboid. And we know that the volume of the cuboid is its length times its width times its height. Next, we worked out the length scale factor of our similar shapes because we knew the length of two corresponding sides. Then, we could use our knowledge of length area and volume scale factors — and in particular the fact that the volume scale factor is the cube of the length scale factor — to work out the volume of the small cuboid. Then, we could take the volume of the small cuboid away from the volume of the large cuboid to work out the remaining volume. And finally, we had to convert a top heavy fraction into a mixed number to check our answer.

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