Video: Finding the Base Radius of a Cone Using Similarity

A pair of cones are similar. Cone A has a base radius of 17 cm and a slant height of 20 cm, whereas cone B has a base radius of 𝑥 cm and a slant height of 48 cm. Determine the value of 𝑥.

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Video Transcript

A pair of cones are similar. Cone A has a base radius of 17 centimetres and a slant height of 20 centimetres, whereas cone B has a base radius of 𝑥 centimetres and a slant height of 48 centimetres. Determine the value of 𝑥.

We’re told, first of all, that this pair of cones are similar. Now, two solids are similar if one is just an enlargement of the other, which means that corresponding lengths on the two cones must be in the same ratio. So, for example, if the radius of one cone is double the radius of the other, then the height of that cone must also be double the height of the other.

We’ve been given some information in the question about some lengths on cone A and cone B. Cone A, first of all, has a base radius of 17 centimetres. That’s the radius of the circle on the base of the cone. It also has a slant height of 20 centimetres. That’s this height here, the diagonal height of the cone, rather than the perpendicular height which joins the point or apex of the cone with the centre of the base. For cone B, we’re told that the base radius is 𝑥 centimetres. And this time, the slant height is 48 centimetres.

Now remember, corresponding lengths on these two cones must be in the same ratio in order for the two cones to be similar, which means if we take a length on cone B and divide it by the corresponding length on cone A, we’ll get the same value every time.

So if we divide the base radius of cone B by the base radius of cone A, that’s 𝑥 over 17, we get the same value as if we divide the slant height of cone B by the slant height of cone A. That’s 48 over 20. So by equating these two ratios, we have an equation that we can solve in order to find the value of 𝑥.

To find 𝑥, we need to multiply both sides of this equation by 17, giving 𝑥 is equal to 17 multiplied by 48 over 20. Evaluating this on a calculator gives 40.8. We don’t need to include units for 𝑥 because we’re told that the base radius of cone B is 𝑥 centimetres. So the units are already included.

The value of 𝑥 then is 40.8.

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