Question

If the ratio of the volume of two similar solids is 8:125, then what is the ratio of their surface areas?

Answer 1

lets take a rectangular solid, in the form of a long prism.

V=bhl
Area= 2(bh+lh+bl)

let l= 8, b=8h
so, for v=8, l=8, b,h=1
now scale it up to 125/8
l=8*cubrt(125/8)=8*5/2=40
b= 5/2 h=5/2
original surface area: 2(1+8+8)=34
new surface area: 2(25/4+200/2+200/2)
= 2(412.5/2)=412.3

ratios of surefce area 34:412.3

Now you need to figure other shapes. Try a cube, and a sphere.

Answer 2

If two solids have volume ratio r^3, then

their linear ratio (sides length) is r
their area ratio (surface area) is r^2

So, since
v/V = (2/5)^3
a/A = (2/5)^2 = 4/25

You gained a factor of 2 there when you said 8*5/2 = 40