Video: AQA GCSE Mathematics Higher Tier Pack 4 • Paper 3 • Question 25

The volume of a cone = (1/3) 𝜋𝑟²ℎ. The cone whose base has center 𝑂 has a perpendicular height of eight centimeters and a volume of (32/3) 𝜋. 𝐵𝑂 is perpendicular to 𝑂𝐴. Calculate the angle between the base and 𝐴𝐵.

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Video Transcript

The volume of a cone equals a third 𝜋𝑟 squared ℎ. The cone whose base has center 𝑂 has a perpendicular height of eight centimeters and a volume of 32 over three 𝜋. 𝐵𝑂 is perpendicular to 𝑂𝐴. Calculate the angle between the base and 𝐴𝐵.

So first of all, what we’re going to do is use the information that we’ve got from the question and see if we can find any missing values. So we’ve got the volume is equal to 32 over three 𝜋. And we know that the volume of a cone is equal to a third 𝜋𝑟 squared ℎ. So therefore, we know that 32 over three 𝜋 is equal to a third 𝜋𝑟 squared ℎ, cause that’s our volume.

Well, we know that the perpendicular height is eight centimeters. So we can substitute that in for ℎ. Therefore, we can rearrange and solve to find 𝑟. So when we substitute ℎ equals eight in, we’re gonna get 32 over three 𝜋 is equal to a third 𝜋𝑟 squared multiplied by eight.

First of all, we can see that we’ve got some similarities between each side of the equation. We’ve got a fraction that’s got three as a denominator, and we’ve got 𝜋. So therefore, we can actually cancel these by multiplying each side of the equation by three and dividing through by 𝜋. So therefore, we’re left with 32 is equal to eight 𝑟 squared.

And then if we divide both sides of the equation by eight, we’re gonna get four is equal to 𝑟 squared. So then if we take the square root of both sides of the equation, cause we’ve got 𝑟 squared, we’re gonna get 𝑟 is equal to two. I flipped around here just so we’ve got the 𝑟 on the left-hand side. We’re only interested in the positive answer because we’re dealing with a length. So it’ll be positive. So we know that the radius is equal to two.

So now what we can do is we can use this and the other values that we know to find the angle between the base and 𝐴𝐵, which I’ve marked on our diagram as 𝜃. So what do we do next? Well, we see that we’re actually dealing with a right-angled triangle, because we’re told that 𝐵𝑂 is perpendicular to 𝑂𝐴. So that’s going to be a right angle between those two lines. So what I’ve done is I’ve drawn the right-angled triangle. So we can see it nice and clearly.

Now because we’ve got a right-angled triangle, we can use a couple of things. We could use Pythagoras’s theorem. But as we’re trying to find an angle, this would not be suitable. So therefore, we can use the trig ratios or trigonometric ratios. And these are sine, cosine, and tangent or sin, cos, and tan. And we’ve got a memory aid that helps us decide which one to use. And that’s SOHCAHTOA.

Okay, so we’re gonna use our trig ratios to find the angle 𝜃. So what do we do? Well, I like to do it in certain steps. And the first step is to label the sides of our triangle. So first, we have the hypotenuse, which is the longest side and opposite the right angle. Then we have the opposite, which is the side that’s opposite our angle 𝜃. And then we have adjacent, which is the side next to the angle that we’ve got, 𝜃, and between that and the right angle. So that’s step one, label, complete.

Now step two is we need to decide which ratio to use. Well, to enable us to do this, what we do is we look at which side we’ve got. So we’ve got the opposite and we’ve got the adjacent. And 𝑂 and 𝐴 for opposite and adjacent are in TOA. So therefore, we know that we’re gonna use the tan or tangent ratio. So we know that tan 𝜃 is gonna be equal to the opposite divided by the adjacent. So that’s step two complete. We’ve chosen our ratio.

And then step three is substitute. And what I mean by this is substitute in the values that we know. So to do that, what we do is we put in the values we know, which is the opposite, which is eight, and the adjacent, which is two. So we have tan 𝜃 is equal to eight over two. So that’s step three complete. We substituted.

And then, finally, what we’ve got is step four, and that’s solve. So what we need to do now is solve to find 𝜃. Well, first of all, we know that eight over two or eight divided by two is four. So we can say that tan 𝜃 is equal to four.

So now what we want to do is find the inverse tan of each side of the equation. And the inverse tan is tan to the minus one. So if you look on your calculator, it’ll usually be where the tan button is. And you have to press shift and then press the tan button and you’ll get tan to the minus one. What this does is this will enable us to find the angle.

So therefore, we get 𝜃 is equal to 75.96 degrees to two decimal places. And that’s what you get if you put tan to the minus one or inverse tan into your calculator and then four. Well, in some calculators, you put it in the other way around. You’ll have to check what particular calculator you’re using.

So we can say that the angle between the base and 𝐴𝐵 is 75.96 degrees to two decimal places. And we’ve completed the final step, which was solve.

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