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Cone A and cone B are mathematically similar - Edexcel - GCSE Maths - Question 14 - 2017 - Paper 3

Question 14

Cone A and cone B are mathematically similar. The ratio of the volume of cone A to the volume of cone B is 27 : 8. The surface area of cone A is 297 cm². Show that t... show full transcript

Worked Solution & Example Answer:Cone A and cone B are mathematically similar - Edexcel - GCSE Maths - Question 14 - 2017 - Paper 3

Step 1

The ratio of the volumes

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Answer

Given that the ratio of the volumes is ( \frac{V_A}{V_B} = \frac{27}{8} ), we can express the ratio of the linear dimensions of the cones using the formula:

$\frac{V_A}{V_B} = \left(\frac{r_A}{r_B}\right)^3$

From this, we can find the ratio of their radii:

$\frac{r_A}{r_B} = \left(\frac{27}{8}\right)^{\frac{1}{3}} = \frac{3}{2}$

Step 2

The surface area of the cones

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Answer

Since the cones are similar, the ratio of their surface areas can be determined using the square of the ratio of their linear dimensions:

$\frac{S_A}{S_B} = \left(\frac{r_A}{r_B}\right)^2$

So,

$\frac{S_A}{S_B} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$

Now, substituting in the known surface area of cone A:

$\frac{297}{S_B} = \frac{9}{4}$

Step 3

Finding the surface area of cone B

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Answer

Cross-multiplying gives us:

$297 \cdot 4 = 9 \cdot S_B$

which simplifies to:

$1188 = 9 S_B$

Thus,

$S_B = \frac{1188}{9} = 132 cm²$

Therefore, we have shown that the surface area of cone B is indeed 132 cm².

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