Geometry Question?

Question

The masses of two similar objects are 24 kg and 81 kg respectively. If the surface area of the larger object is 540 cm2, find the surface area of the smaller object.

Ans = 240 cm2. Show how to work it out.

Answer 1

volume = mass/density

since the objects are "similar" (in content, shape ...) we can assume that density=1 and hence volumes are 24 units and 81 units

area is square of linar units...while volume is cubic

hence the ratio would be:

24^1/3:81^1/3 = x^1/2 : 540 ^ 1/2

or, x^1/2 = (24/81)^1/3 * 540^1/2 = (8/27)^1/3 * 540^1/2
= 2/3 * 540 ^ 1/2 = (540*4/9)^1/2

of x=540*4/9 = 240 sq.units

note: a similar question (homework!) : if the eiffel tower measures 100 m and weighs 1000 tons. what would an exact model of it, measuing 1m weigh?

Answer 2

Suppose that "similar" means the same density, and the same shape. Think of gradually magnifying the 24 kg object. When you had made it twice as big, it would be eight times the volume and mass. What about its surface area? If it was a box, each side would be twice as wide and twice as long, so four times the surface area. Even when it's not a box, the same rule applies.

So out with the calculator, cube root of (81/24) is 1.5 exactly, that's the factor the larger object has grown by compared to the smaller one. Calculator again, 540 divided by the square of 1.5 is 240. Check the result: scaling factor is 1.5, area scaling factor is 1.5 squared which turns 240 into 540, volume scaling factor is 1.5 cubed which turns 24 into 81.

Answer 3

Let the objects are spheres
V1 = Volume of 1st
V2 = volume of 2nd
M1= mass of 1st
M2= mass of 2nd
R1 =Radius of 1st
R2 =Radius of 2nd
Since masses are directly proportional to their volumes
M1/M2 = V1/V2
24/81=(R1)^3/(R2)^3
8/27 =(R1/R2)^3
2/3=R1/R2
S1= surface area of 1st
S2 =surface area of 2nd
Since surface are are proportional to square of their radius
(2/3)^2 =S1/S2=S1/540
4/9 =S1/540
S1=240 cm^2

Answer 4

the objects are similar
V1/V2=cube root of 24/81=2:3 since the objects are similar

Answer 5

For starters, your question is a lilttle ambiguous in itself.
So, we Satrt with two assumptions:
1)The densities of the two objects are equal (for obvious reasons)
2)The objects have similar shapes but vary in sizes.

so you have m1,m2, sa1, sa2 involved iin equations and p1 and p2 are the variables distinguishing between the two objects.

m1=K(p1)^3, m2=K(p2)^3. => (m1/m2)^(1/3)=p1/p2

sa1=k(p1)^2, sa2=k(p2)^2 => sa1/sa2 = (p1/p2)^2
sa1/sa2 = (m1/m2)^[(1/3)*2]
= (24/81)^(2/3)
therefore,

sa1= (4/9)*540
which equals 240cm2 as required