# 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.