An open box is to be constructed
for a piece of cardboard, eight inches by eight inches, by cutting squares from each
corner and then folding up the sides. Given that the removed squares have
a side length of 𝑥 inches, express the volume of the box in terms of 𝑥 as a
polynomial in standard form.
The first thing I’ve done to
actually help me solve this problem is draw a little sketch. So I’ve got the
eight-inches-by-eight-inches piece of cardboard. Now I’ve added the four corners
that have been cut out. And these, we know from the
question, they are squares and they have side length of eight inches.
Okay, great! So now what else can we do? Well, we want to know this
length. But how can we derive this? Well can say that it’s eight
inches. And that’s because that’s the size
of piece of cardboard minus 𝑥, because that’s the length of one of the squares of
the corner that we had taken off. But then we’ve also got one at the
other end. So it’ll be minus another 𝑥 so
it’s eight minus 𝑥 minus 𝑥.
So this will give us the length of
eight minus two 𝑥. Great! So we now have all the dimensions
we need so we can get on and solve the problem, using this really useful
diagram. But we know that the volume of a
cuboid is equal to the length multiplied by the width multiplied by the height.
So in order to use this formula,
what we’re gonna have to do is actually determine the length, width, and height of
our cuboid. Well first of all, we’re actually
gonna look at the height because if we fold up this side, where we’ve cut the square
from, we can actually see that actually our height is going to be 𝑥 inches.
So we’ll say that the height is
equal to 𝑥. And to help you visualize it I’ve
just gonna draw a quick sketch here cause actually when we fold up the sides, that’s
where the 𝑥 would be.
So that’s what ℎ is. ℎ is equal to 𝑥. Well next, we’re gonna look at the
length and the width because actually because it’s cut from a square piece of
cardboard, which is eight inches by eight inches, they’re actually going to be the
So they’re both gonna be eight
minus two 𝑥. And again I’ve shown them on my
rough sketch here. And that’s because actually when
you fold up the sides, that’s the base that we’re gonna be left with is a base which
is eight minus two 𝑥 by eight minus two 𝑥.
Great! So now we’ve got our dimensions in
terms of 𝑥, we can actually go about and find our volume in terms of 𝑥 as a
polynomial. So when we substitute in our terms,
we get the volume is equal to 𝑥 multiplied by eight minus two 𝑥 multiplied by
eight minus two 𝑥. I’ve just put the 𝑥 at the
beginning just because it’s gonna help us in the next step.
Now the next step in simplifying is
actually we’re gonna expand the first parentheses by multiplying it by 𝑥. So as I’ve shown, we start with 𝑥
multiplied by eight, this is gonna give us eight 𝑥, and then 𝑥 multiplied by
negative two 𝑥, which is gonna give us negative two 𝑥 squared. So we’ve now got our first
parentheses, which is eight 𝑥 minus two 𝑥 squared. And then this is multiplied by
eight minus two 𝑥.
Our next step is to actually expand
the parentheses. So we have eight 𝑥 multiplied by
eight, which gives us 64𝑥, and then eight 𝑥 multiplied by negative two 𝑥. That gives us negative 16𝑥
squared. So that’s great! We’ve now multiplied the first term
in our first parentheses by both terms in our second parentheses.
Now we’re gonna do the second term
in our first parentheses. So we have negative two 𝑥 squared
multiplied by eight, which gives us negative 16𝑥 squared. And then finally, we’ve got
negative two 𝑥 squared multiplied by negative two 𝑥 which gives us plus four 𝑥
Brilliant! So we’ve now got the expression
which is 64𝑥 minus 16𝑥 squared minus 16𝑥 squared plus four 𝑥 cubed. Okay, fantastic! We’re on to our last stage. And that is we need to
simplify. Okay, so first we have four 𝑥
cubed. And the reason we have that is
because we like to start with the highest order of 𝑥 first.
Now we can say that it’s four 𝑥
cubed minus 32𝑥 squared. And it’s minus 32𝑥 squared because
we’ve got negative 16𝑥 squared minus 16𝑥 squared. So that gives us minus 32𝑥
squared. And finally, we have plus 64𝑥. So therefore, we can say that the
volume of the box, in terms of 𝑥 as a polynomial in standard form, is equal to four
𝑥 cubed minus 32𝑥 squared plus 64𝑥.