Question

11. if the scale factor of two similar solids in 3 : 14, what is the ratio of their corresponding areas? what is the ratio of their corresponding volumes?

Answer 1

Part a: Ratio between areas:
We are given that the ration between the sides of similar solid is 3:14
The ration between the areas can be obtained by simply squaring the ratio between the sides (raising the ration to the power of 2)
This is because the ratio between the sides is only linear. We must square it to get the area ratio
Therefore:
ratio between areas = (3)² : (14)²
ratio between areas = 9 : 196

Part (b): Ratio between volumes:
We are given that the ration between the sides of similar solid is 3:14
The ration between the volumes can be obtained by simply cubing the ratio between the sides (raising the ration to the power of 3)
This is because the ratio between the sides is only linear. We must cube it to get the volume ratio
Therefore:
ratio between volumes = (3)³ : (14)³
ratio between volumes = 27 : 2744

Hope this helps :)

Answer 2

For this case we have the following scale factor:
k = 3/14
Therefore, for the relationship of areas we have:
A1 / A2 = k ^ 2 = (3/14) ^ 2
A1 / A2 = 9/196
For the relation of volumes we have:
V1 / V2 = k ^ 3 = (3/14) ^ 3
V1 / V2 = 27/2744
Answer:
The ratio of their corresponding areas is:
9: 196
The ratio of their corresponding volumes is:
27: 2744