Video: Pack 1 โ€ข Paper 1 โ€ข Question 15

Pack 1 โ€ข Paper 1 โ€ข Question 15

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Video Transcript

A sphere has a volume of 4000 centimeters cubed. Part a) by estimating ๐œ‹ 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 ๐œ‹, 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 weโ€™re trying to find. So first of all, we know that our volume is equal to 4000 โ€” so 4000 centimeters cubed. Okay, well then next, what we can actually say is we know what ๐œ‹ is because in the question it tells us to estimate ๐œ‹ to one significant figure. So letโ€™s take a look at ๐œ‹.

So we know that ๐œ‹ 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 ๐œ‹ is equal to three for the purposes of our question.

Okay, great, then, finally, we have ๐‘Ÿ. But ๐‘Ÿ is what we are trying to find out. So great, we now know the information that weโ€™ve been given by the question. Then, next, what weโ€™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 ๐œ‹๐‘Ÿ 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 ๐‘Ÿ cubed. And thatโ€™s because 4000 was our volume and three was our estimation for ๐œ‹.

Okay, great, so now if we actually take a look at how weโ€™d simplify this and start to solve. The first thing we can do is actually cancel threes here because weโ€™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 letโ€™s rewrite this. So weโ€™re now left with 4000 is equal to four ๐‘Ÿ cubed. So then, the next stage is divide everything by four. So weโ€™re dividing both sides of our equation by four. So weโ€™re gonna get 1000 is equal to ๐‘Ÿ cubed. So then if we take the cube root of each side, we get the cube root of 1000 is equal to ๐‘Ÿ. So therefore, we can say that ๐‘Ÿ โ€” so our radius โ€” is gonna be equal to 10 centimeters.

Okay, great, so weโ€™ve actually finished part a. Letโ€™s move on to part b. Well, for part b, what weโ€™re actually looking at is how a more accurate approximation for ๐œ‹ would actually affect the radius. Well, letโ€™s have a look. So weโ€™re gonna start by thinking about the ๐œ‹ that we looked at earlier. So we say that ๐œ‹ 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, letโ€™s look at two significant figures and three significant figures.

Well, if we actually looked at two significant figures, then itโ€™ll be 3.1. And thatโ€™s because our second significant figure is one. And again, the number after is a four. So itโ€™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 โ€” so theyโ€™re both more accurate approximations โ€” 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 ๐œ‹ will increase as weโ€™ve shown. So therefore, ๐‘Ÿ must decrease.

But hold on! How have I suddenly made this assumption that ๐‘Ÿ must decrease? Well, if we look back at our equation, we can see that we know our volume โ€” thatโ€™s 4000. And if our volume is equal to four-thirds multiplied by ๐œ‹ multiplied by ๐‘Ÿ cubed, if our ๐œ‹ is actually increasing, then to reach the same amount of volume to 4000, then therefore, our ๐‘Ÿ โ€” so our radius โ€” must decrease.

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