Photo AI
Three solid shapes A, B and C are similar - Edexcel - GCSE Maths - Question 16 - 2018 - Paper 1
Question 16
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 vol... show full transcript
Worked Solution & Example Answer:Three solid shapes A, B and C are similar - Edexcel - GCSE Maths - Question 16 - 2018 - Paper 1
Step 1
Work out the ratio of length A to length B
Answer
Since the shapes are similar, the ratio of the surface areas of shapes is equal to the square of the ratio of their corresponding lengths. Let the ratio of lengths A to B be k.
We have:
[ \frac{4}{25} = k^2 ]
To find k, take the square root of both sides:
[ k = \sqrt{\frac{4}{25}} = \frac{2}{5} ]
Therefore, the ratio of the lengths of shapes A and B is 2:5.
Step 2
Work out the ratio of length C to length B
Answer
For the ratio of lengths B to C, we can derive this from the volume ratio given.
The volumes of similar shapes are in the ratio of the cubes of their corresponding lengths. Given: [ \frac{27}{64} = \left( \frac{b}{c} \right)^3 ]
Taking the cube root gives: [ \frac{b}{c} = \sqrt[3]{\frac{27}{64}} = \frac{3}{4} ]
Thus, the ratio of lengths B to C is 3:4.
Step 3
Calculate the ratio of height A to height C
Answer
Since we have previously established the ratios:
- Length A to Length B is 2:5
- Length B to Length C is 3:4
To find the ratio of height A to height C, we need to find the ratio of Length A to Length C:
Using the known ratios: [ \frac{A}{C} = \frac{A}{B} \times \frac{B}{C} = \frac{2}{5} \times \frac{5}{4} = \frac{2}{4} = \frac{1}{2}]
Thus, the ratio of height A to height C is 1:2.