Bradley's Maths
Similar Solids: Linear, Area, and Volume Scale
GCSE all examination boards and IGCSE Cambridge (0580)
Exam Question
Two statues, A and B are mathematically similar.
A has a surface area of 350 cm$^2$.
B has a surface area of 3150 cm$^2$.
a) Given that the height of statue A is 12 cm, find the height of statue B.
b) Given that the volume of statue B is 5184 cm$^3$, Find the volume of statue A.
Before you move on to the solution or watch the video, why don't you have a go at this one yourself?
Solution
This question is a mid-exam calculator question, so GCSE Paper 2 around question 10 to 15 and IGCSE Paper 4 in about the same place. Getting the full marks depends on you knowing the relationship between lengths and volumes in similar solids:
$$\text{Length} : \text{Area} : \text{Volume}$$ $$ k : k^2 : k^3 $$a) We need to find $k^2$, the area scale factor and we do that by dividing the surface area of statue B by the surface area of statue A.
$$k^2 = \dfrac{3125}{350} = 9$$We take the square root of this to get the linear scale factor.
$$k = \sqrt{9} = 3$$And we multiply this by the height of A to get the height of B.
$$Height_B = 12 \times 3 = 36cm$$b) Here we are given the volume of statue B and need to find the volume of Statue A. The volume scale factor is the cube of the linear scale factor
$$\text{Volume scale factor }= k^3 = 3^3 = 27$$We are going from the larger statue to the smaller statue so we must divide by the scale factor.
$$Vol_A = \dfrac{5184}{27} = 192cm^3$$You should try to learn the relationship between length, area, and volume:
$$\text{Length} : \text{Area} : \text{Volume}$$ $$ k : k^2 : k^3 $$Doing so is the key to answering these questions!
Is there a topic you'd like to see featured? Request it on my Q & A page and I'll prioritize it for a future post.
Topics covered in this question
- Using the linear:area:volume = $k : k^2 : k^3$ scale
- Finding an unknown length given both areas and a known length
- Finding an unknown volume given two lengths and a known volume
- Working from a smaller item to a larger and vice-versa