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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

We know that for similar shapes, the ratio of their volumes is equal to the cube of the ratio of their corresponding linear dimensions. Given the volume ratio of cone A to cone B as 27:8, we can express this as:

$\frac{V_A}{V_B} = \frac{27}{8}$

This implies that the ratio of their linear dimensions (heights and radii) is:

$\frac{h_A}{h_B} = \sqrt[3]{\frac{27}{8}} = \frac{3}{2}$

Step 2

Finding the ratio of the surface areas

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Answer

The ratio of surface areas of two similar shapes is equal to the square of the ratio of their corresponding linear dimensions. Therefore:

$\frac{SA_A}{SA_B} = \left(\frac{h_A}{h_B}\right)^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$

Step 3

Calculating the surface area of cone B

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Answer

Now, we can use the surface area of cone A to find that of cone B:

$\frac{SA_A}{SA_B} = \frac{9}{4} \implies SA_B = SA_A \times \frac{4}{9}$

Substituting the value of ( SA_A = 297 \ cm^2 ):

$SA_B = 297 \times \frac{4}{9} = 132 \ cm^2$

Thus, the surface area of cone B is confirmed to be 132 cm².

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