If the ratio of the volume of two similar solids is 8:125, then what is the ratio of their surface areas?
To find the ratio of the surface areas of two similar solids, we can use the fact that the ratio of their volumes is equal to the cube of the ratio of their corresponding side lengths. The surface area of a solid is directly proportional to the square of its side length.
Given that the ratio of the volume of the two similar solids is 8:125, we can find the cube root of each ratio to determine the ratio of their side lengths:
Cube root of 8 = 2
Cube root of 125 = 5
Therefore, the ratio of their side lengths is 2:5.
Since the surface area is directly proportional to the square of the side length, we can square the ratio of their side lengths to find the ratio of their surface areas:
(2^2):(5^2)
4:25
So, the ratio of their surface areas is 4:25.
To find the ratio of the surface areas of two similar solids, we need to understand the relationship between their volumes and surface areas.
For similar solids, the ratio of their volumes is equal to the cube of the ratio of their corresponding side lengths. In other words:
(Volume of first solid) / (Volume of second solid) = (Side length of first solid / Side length of second solid)^3
In this case, the given ratio of volumes is 8:125. This means the cube of the ratio of the corresponding side lengths will also be 8:125. To find the side length ratio, we can take the cube root of the volume ratio:
(Cube root of 8) : (Cube root of 125)
The cube root of 8 is 2, and the cube root of 125 is 5. So, the ratio of the side lengths is 2:5.
Now, to find the ratio of surface areas, we can square the ratio of the corresponding side lengths. Therefore:
(Surface area of first solid) / (Surface area of second solid) = (Side length of first solid / Side length of second solid)^2
Plugging in the side length ratio of 2:5, we have:
(2:5)^2 = (2^2) : (5^2) = 4:25
Therefore, the ratio of the surface areas of the two similar solids is 4:25.
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
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.