The answer to this is 1:9, however I had 1:27. The solutions said the scale factor had to be squared rather then cube because the length wasn't changing as the cross-sectional area did or something like that..I can't really follow what they mean...any help would be appreciated thanks!
Let's simplify the question by considering a cube with side length L. The volume is LxLxL =L
3If we triple the length of
only one side, the volume is now LxLx3L=3L
3 . This is only triple the original volume.
If we were to triple
two of the sides of the original box, the volume is then Lx3Lx3L = 9L
3 or 3
2L
3Only when we triple
all three of the original sides will be get 3Lx3Lx3L = 27L
3 which is 3
3 times the original volume
In summary - tripling only one dimension (ie L ot W or H) will multiply the original volume by 3 - ie only multiply by one scale factor
Tripling two dimensions only will multiply the original volume by 3
2 - ie square the scale factor
Tripling all three dimensions will multiply the original volume by 3
3 - ie cube the scale factor
Similarly if you were to multiply any two dimensions by 7, the volume will increase by a factor of 7
2 or 49
In your example, the ratio of the height of water:height of gauge = 1:3 or 1/3 as a fraction. This would also mean that the distance across the surface of the water to the sloping edge:similar distance across the top of the guage is also 1:3 or 1/3 as a fraction. But the width of the rain guage (between paralell triangular faces) has not changed. So only two dimensions are different and so we only square two ratios as in (1/3)
2 = 1/9 or 1:9