7. Variations

Two quantities are said to be vary as with each other if the ratio between them is constant. For example, if \(a=kb\), where \(a\) and \(b\) are variables and \(k\) is a constant which is also known as proportionality constant. Most of the time it is said that \(a\) is directly proportional to \(b\) or \(a\) varies directly as \(b\).

The symbol \(\varpropto\) is used to denote variation; so that \(a\varpropto b\) is read as “\(a\) varies as \(b\)”.

For example, if a person drink 5 liters of water in one day then in 10 days he will drink 50 liters and in 20 days he will drink 100 liters.

One quantity \(a\) is said to vary inversely as another \(b\) or \(a\) is inversely proportional to \(b\) if product of these two is constant.

Therefore, \(ab=k\) or \(a=\frac{k}{b}\).

For example, if \(a\) persons do a work in \(b\) days then 1 person will do it in \(\frac{b}{a}\) days. So our proportionality constant is \(\frac{b}{a}\).

One quantity is said to vary jointly as a number of others, when it varies directly as their product.

This \(a\) varies jointly as \(b\) and \(c\), when \(a=kbc\). For example, second law of Newton says force is equal to mass multiplied by acceleration or \(F=ma\).

7.1. Problems

  1. If \(x\) varies as \(y\), and \(x=8\) when \(y=15\), find \(x\) when \(y=10\).
  2. If \(x\) varies inversely as \(y\), and \(x=7\) when \(y=3\), find \(x\) when \(y=7\).
  3. If the square of \(x\) varies as the cube of \(y\), and \(x=3\) when \(y=4\) find the value of \(y\) when \(x=\frac{1}{\sqrt{3}}\).
  4. \(x\) varies as \(y\) and \(z\) jointly; if \(x=2\) when \(y=\frac{3}{5}\) and \(z=\frac{10}{27}\) find \(z\) when \(x=54\) and \(y=3\).
  5. If \(a\) varies as \(c\), and \(b\) varies as \(c\), then \(a\pm b\) and \(\sqrt{ab}\) will each vary as \(c\).
  6. If \(a\) varies as \(bc\), then \(b\) varies inversely as \(\frac{c}{a}\).
  7. \(a\) varies directly as \(b\) and inversely as \(c\); also \(a=\frac{2}{3}\) when \(b=\frac{3}{7}\) and \(c=\frac{9}{14}\); find \(b\) when \(a=\sqrt{48}\) and \(c=\sqrt{75}\).
  8. If \(x\) varies as \(y\), prove that \(x^2+y^2\) varies as \(x^2-y^2\).
  9. If \(y\) varies as the sum of two quantities, one of which varies directly as \(x\) and the other inversely as \(x\); and if \(y=6\) when \(x=4\), and \(y=\frac{10}{3}\) when \(x=3\); find the equation between \(x\) and \(y\).
  10. If \(y\) is equal to the sum of two quantities, one of which varies as \(x\) directly, and the other as \(x^2\) inversely; and if \(y=19\) when \(x=2\), or \(3\); find \(y\) in terms of \(x\).
  11. If \(a\) varies directly as the square root of \(b\) and inversely as the cube of \(c\), and if \(a=3\) when \(b=256\) and \(c=2\), find \(b\) when \(a=24\) and \(c=\frac{1}{2}\).
  12. Given that \(x+y\) varies as \(z+\frac{1}{z}\), and that \(x-y\) varies as \(z-\frac{1}{z}\), find the relation between \(x\) and \(z\), provided that \(z=2\) when \(x=3\) and \(y=1\).
  13. If \(a\) varies as \(b\) and \(c\) jointly, while \(b\) varies as \(d^2\), and \(c\) varies inversely as \(a\), show that \(a\) varies as \(d\).
  14. If \(y\) varies as the sum of three quantities of which the first is constant, the second varies as \(x\), and the third as \(x^2\); and if \(y=0\) when \(x=1, y=1\) when \(x=2\), and \(y=4\) when \(x=3\); find \(y\) when \(x=7\).
  15. When a body falls from rest its distance from the resting point varies as the square of the time it has been falling: if a body falls though \(402\frac{1}{2}\) feet in 5 seconds, how far does it fall in 10 seconds? Also how far does it fall in the 10th second?
  16. Given that the volume of a sphere varies as the cube of its radius and that when the radius is \(3\frac{1}{2}\) feet the volume is \(179\frac{2}{3}\) cubic feet, find the volume when the radius is 1 foot 9 inches.