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Three solid shapes A, B and C are similar - Edexcel - GCSE Maths - Question 15 - 2018 - Paper 1
Question 15
Three solid shapes A, B and C are similar. The surface area of shape A is 4 cm². The surface area of shape B is 25 cm². The ratio of the volume of shape B to the v... show full transcript
Worked Solution & Example Answer:Three solid shapes A, B and C are similar - Edexcel - GCSE Maths - Question 15 - 2018 - Paper 1
Step 1
Work out the ratio of the surface areas of shapes A and B
Answer
The ratio of the surface areas of similar shapes is proportional to the square of their corresponding lengths.
Let the ratio of the lengths of shapes A and B be denoted as ( k ).
- Surface area of shape A: ( SA_A = 4 \text{ cm}^2 )
- Surface area of shape B: ( SA_B = 25 \text{ cm}^2 )
The ratio can be expressed as:
[ \frac{SA_A}{SA_B} = \frac{4}{25} = \left( \frac{l_A}{l_B} \right)^2 ]
So, ( \frac{l_A}{l_B} = \sqrt{\frac{4}{25}} = \frac{2}{5} ).
Step 2
Use the ratio of the volumes to find the ratio of lengths
Answer
The ratio of the volumes of similar shapes is proportional to the cube of their lengths.
Given that the ratio of volumes of shape B to shape C is ( \frac{V_B}{V_C} = \frac{27}{64} ):
This implies: [ \frac{l_B}{l_C} = \sqrt[3]{\frac{27}{64}} = \frac{3}{4} ]
Step 3
Set up the ratios and solve for height
Answer
Using the ratios obtained:
-
From ( \frac{l_A}{l_B} = \frac{2}{5} ):
[ l_A = \frac{2}{5} l_B ] -
From ( \frac{l_B}{l_C} = \frac{3}{4} ):
[ l_B = \frac{3}{4} l_C ]
Substitute ( l_B ) into the equation for ( l_A ): [ l_A = \frac{2}{5} \left(\frac{3}{4} l_C \right) = \frac{6}{20} l_C = \frac{3}{10} l_C ]
Thus, the ratio of the height of shape A to the height of shape C is: [ \frac{h_A}{h_C} = \frac{3}{10} ] and can be expressed as ( 3:10 ).