Video Transcript
Consider the square prism shown in
the diagram. Write its surface-area-to-volume
ratio in terms of 𝑥. Give your answer in standard
form.
And there’s also a second part to
this question which we’ll deal with later. It’s quite clear that to answer
this question, we’re going to need to form expressions for both the surface area and
the volume of the square prism. Remember, the surface area of a
three-dimensional shape is the combined area of all its faces. We could say that our square prism
is a cuboid. And cuboids have six rectangular,
or sometimes square, faces. So, it’s really sensible to be
methodical about how we calculate each of their areas.
Let’s begin by forming expressions
for the area of the rectangle at the front of our diagram. The formula for area of a rectangle
is its base multiplied by its height. So, the area of this rectangle can
be found by multiplying one plus two 𝑥 by one plus 𝑥.
Now spot that we’ve been asked to
give our answer in standard form. This is not to be confused with the
standard form that we often use to describe numbers which are very very small or
very very large. When we write an expression in
standard form, it’s important that each variable has one factor. So, we can’t have any brackets in
there. And we write these in descending
powers. So, in this example, we’re going to
write it in descending powers of 𝑥.
So, lets expand these brackets as
normal. We could use the grid method or the
FOIL. method. Let’s look at the FOIL. method for
expanding these brackets. F stands for first. We multiply the first term in the
first bracket by the first term in the second bracket. One multiplied by one is one. O stands for outer. We multiply the outer term in each
bracket. That gives us 𝑥. I stands for inner. We multiply the inner terms. And two 𝑥 multiplied by one is
just two 𝑥. And L stands for last. We multiply the last term in each
bracket. Two 𝑥 multiplied by 𝑥 is two 𝑥
squared.
And all that’s left to do is to
simplify this expression by collecting like terms and write it in descending powers
of 𝑥. That’s from largest to
smallest. 𝑥 plus two 𝑥 is three 𝑥. So, we can write the expression for
the area of this rectangular face as to 𝑥 squared plus three 𝑥 plus one.
But this is a prism. So, this means the area of the face
on the opposite side, that’s kind of the back of the prism, if you will, is the
same. And, in fact, since the prism has a
square base, we can see that the rectangular face to the right of our shape also has
an area of two 𝑥 squared plus three 𝑥 plus one as does the face directly opposite
that. That’s the face on the left-hand
side of our prism.
This means we have four faces with
an area of two 𝑥 squared plus three 𝑥 plus one. And we can expand these brackets by
multiplying each term by four. And we see that the area of the
four rectangular faces is eight 𝑥 squared plus 12𝑥 plus four.
And next we have the face at the
top of our prism. It has a base of one plus two 𝑥
and a height of one plus two 𝑥. So, its area is one plus two 𝑥 all
squared, or one plus two 𝑥 multiplied by one plus two 𝑥. And expanding as we did before, or
distributing these brackets using the FOIL method, we get one plus two 𝑥 plus two
𝑥 plus four 𝑥 squared, which in standard form is four 𝑥 squared plus four 𝑥 plus
one.
But how many other faces have this
area? Well, we have an identical face at
the bottom of our prism. So, there are two faces with an
area of four 𝑥 squared plus four 𝑥 plus one. And distributing these brackets by
multiplying each term by two, we get the total area of these two faces to be eight
𝑥 squared plus eight 𝑥 plus two.
Remember, we said that the surface
area is the combined area of all six faces on our square prism. So, that’s eight 𝑥 squared plus
12𝑥 plus four plus eight 𝑥 squared plus eight 𝑥 plus two. And since we’re only adding and
adding is commutative, we can see that we don’t really need these brackets. In descending powers of 𝑥, we have
eight 𝑥 squared plus eight 𝑥 squared, which is 16𝑥 squared, 12𝑥 plus eight 𝑥 is
20𝑥, and four plus two is six. So, the total surface area of our
square prism is 16𝑥 squared plus 20𝑥 plus six.
Our next step is to form an
expression for the volume of the square prism. The volume of a prism is found by
multiplying the area of its cross-section by its length. Now we said that the cross section
of this prism is a square. So, that’s its base multiplied by
its height multiplied by its length. And in fact, we can generalise that
the volume for any cuboid is its base multiplied by its height multiplied by its
length.
And in any order, the volume of
this square prism is one plus two 𝑥 multiplied by one plus two 𝑥 multiplied by one
plus 𝑥. Remember, multiplication is
commutative. So, it doesn’t really matter if we
put one plus 𝑥 at the front or in the middle. This is going to give us the same
answer when we distribute these brackets. And to distribute these brackets,
we begin by expanding, or distributing, any two of the brackets. Let’s choose one plus two 𝑥
multiplied by one plus two 𝑥. We already saw that that gives us
four 𝑥 squared plus four 𝑥 plus one.
And to multiply this by one plus
𝑥, the grid method could be a nice way to check that you don’t make any
mistakes. We begin by multiplying four 𝑥
squared by one. That’s four 𝑥 squared. Four 𝑥 multiplied by one is four
𝑥. And one multiplied by one is
one. Then four 𝑥 squared multiplied by
𝑥 is four 𝑥 cubed. Four 𝑥 multiplied by 𝑥 is four 𝑥
squared. And one multiplied by 𝑥 is 𝑥. We have four 𝑥 cubed plus eight 𝑥
squared, that’s four 𝑥 squared plus four 𝑥 squared, plus five 𝑥 plus one.
And all that’s left is to write the
ratio of the surface area to its volume. To achieve this, we divide the
expression we formed for the surface area by the expression we formed for the volume
of the square prism. it’s 16𝑥 squared plus 20𝑥 plus six all over four 𝑥 cubed
plus eight 𝑥 squared plus five 𝑥 plus one. Let’s clear some space and consider
the second part of this question.
The diagram shows the graph of the
surface-area-to-volume ratio of the prism as a function of 𝑥. Which of the following is an
approximate value of 𝑥 for which the surface-area-to-volume ratio is one? Is it A) six, B) 3.3, C) 2.3, D)
1.3, or E) 1.5?
And it’s important to note that
your answers might be in a different order. So, how do we answer this
question? Well, we’re told that the
surface-area-to-volume ratio is one. And if we look at the graph, we can
see that the surface-area-to-volume ratio of the prism is represented by the values
on the 𝑦-axis. The surface-area-to-volume ratio is
one is over here. So, we’re going to add a horizontal
line onto our graph until we hit the curve. From there, we’re going to add a
vertical line downwards until we hit the 𝑥-axis. To work out the value of 𝑥, which
corresponds to a surface-area-to-volume ratio of one, we need to find the scale on
the 𝑥-axis.
We can see that five small squares
on our 𝑥-axis represent one unit. If we divide by five, we then see
that one square is equal to one-fifth of a unit, or 0.2 of a unit. Our value of 𝑥 lies halfway
through a square though, so we divide again by two. And we see that half a square is
equal to 0.1 of a unit. Our value of 𝑥 is one and a half
squares above the number three. So, that’s three plus the value of
one small square, which is 0.2, plus the value of that half a square, which is 0.1,
which is 3.3.
And we see that the approximate
value of 𝑥 for which the surface-area-to-volume ratio is one is 3.3. And we can check this if we
want. We check it by substituting it into
the ratio we formed in the first part of this question. Substituting 𝑥 equals 3.3 into
that formula gives us 0.991, which is extremely close to one, the ratio we were
looking for.
