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
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
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.