Explanation:
x and y are directly proportional means that
y = kx
y grows with the same factor k applied to x.
the area of a circle (the cross section of a cylindrical pipe) is
pi × r²
r being the radius (which is half of the diameter).
so, the area in terms of the diameter is
pi × (d/2)² = pi × d²/4
so, we know
36 correlates to pi × (d small)²/4
60 correlates to pi × (d large)²/4
therefore (directly proportional),
d small² = 36×4/pi
d small = 12/sqrt(pi)
d large² = 60×4/pi
d large = sqrt(4×15×4/pi) = 4×sqrt(15/pi)
the difference between large and small diameter (increase from small to large) is then
4×sqrt(15/pi) - 12/sqrt(pi)
d small = 100%
1% = 100%/100 = 12/sqrt(pi) / 100 = 12/(100sqrt(pi))
the % of the diameter increase is then
increase/1% =
(4×sqrt(15/pi) - 12/sqrt(pi)) / 12/(100sqrt(pi)) =
= (400sqrt(pi)×sqrt(15/pi) - 1200sqrt(pi)/sqrt(pi)) / 12 =
= (400×sqrt(15) - 1200) / 12 = 100×sqrt(15)/3 - 100 =
= 100×(sqrt(15)/3 - 1) = 29.09944487...% ≈ 29.1%
the diameter has to increase by about 29.1%, so that the pipe can carry 60 liters per second.
please consider : only the area of the cross section and the carrying capacity are directly proportional.
the diameter of the cross section and the area of the cross section are NOT directly proportional.
they have the pi×(d/2)² relationship.
and so, while the capacity and the cycle area both increase by the same factor, the impact on the diameter (or radius) is different.
so, for capacity and area we have
60 = k×36
k = 60/36 = 5/3 = 1.66666666...
and therefore both increase by 66.66666...%.
but because of the square relation between diameter and cross section area, the diameter only increases by 29.1%.