A sphere has a volume of 4000
centimeters cubed. Part a) by estimating 𝜋 to one
significant figure, calculate an estimate for the length of the radius of the
sphere. Part b) if you were to calculate
the length of the radius using a more accurate approximation for 𝜋, would this give
you a radius that was longer or shorter than your estimate?
So the way I always start a problem
like this is actually write down the information we know or we’re trying to
find. So first of all, we know that our
volume is equal to 4000 — so 4000 centimeters cubed. Okay, well then next, what we can
actually say is we know what 𝜋 is because in the question it tells us to estimate
𝜋 to one significant figure. So let’s take a look at 𝜋.
So we know that 𝜋 is equal to
3.1415 et cetera. If we want to round it to one
significant figure, so the first significant figure is three, so we look at the
number afterwards, which will actually be a one. And as this is actually below five,
then three would stay as it is. So therefore, we can say that 𝜋 is
equal to three for the purposes of our question.
Okay, great, then, finally, we have
𝑟. But 𝑟 is what we are trying to
find out. So great, we now know the
information that we’ve been given by the question. Then, next, what we’ll need to do
is actually recall one of our volume formulae. And this one is the volume of a
sphere is equal to four-thirds 𝜋𝑟 cubed. Okay, so now what we need to do is
actually substitute the values that we know into our formula. And when we do that, we get 4000 is
equal to four over three multiplied by three multiplied by 𝑟 cubed. And that’s because 4000 was our
volume and three was our estimation for 𝜋.
Okay, great, so now if we actually
take a look at how we’d simplify this and start to solve. The first thing we can do is
actually cancel threes here because we’ve got four-thirds multiplied by three. Well, if you got four-thirds
multiplied by three, that would give us twelve thirds, which is just four. Okay, so let’s rewrite this. So we’re now left with 4000 is
equal to four 𝑟 cubed. So then, the next stage is divide
everything by four. So we’re dividing both sides of our
equation by four. So we’re gonna get 1000 is equal to
𝑟 cubed. So then if we take the cube root of
each side, we get the cube root of 1000 is equal to 𝑟. So therefore, we can say that 𝑟 —
so our radius — is gonna be equal to 10 centimeters.
Okay, great, so we’ve actually
finished part a. Let’s move on to part b. Well, for part b, what we’re
actually looking at is how a more accurate approximation for 𝜋 would actually
affect the radius. Well, let’s have a look. So we’re gonna start by thinking
about the 𝜋 that we looked at earlier. So we say that 𝜋 is equal to
3.1415 et cetera. Well, the question actually asks us
what will happen if we had more accurate approximations. So just as a bit of an example,
let’s look at two significant figures and three significant figures.
Well, if we actually looked at two
significant figures, then it’ll be 3.1. And that’s because our second
significant figure is one. And again, the number after is a
four. So it’s less than five. So it stays as it is. And then, if we take a look at
three significant figures. Well, this is gonna be 3.14.
Well, the key thing is here that
both of these approximations — so they’re both more accurate approximations — are
actually greater than three, so greater than the approximation we used in part
a. So therefore, we can actually say
that a more accurate approximation means that our 𝜋 will increase as we’ve
shown. So therefore, 𝑟 must decrease.
But hold on! How have I suddenly made this
assumption that 𝑟 must decrease? Well, if we look back at our
equation, we can see that we know our volume — that’s 4000. And if our volume is equal to
four-thirds multiplied by 𝜋 multiplied by 𝑟 cubed, if our 𝜋 is actually
increasing, then to reach the same amount of volume to 4000, then therefore, our 𝑟
— so our radius — must decrease.