To say two quantities
and
"vary directly" with one another means that a change in one of the quantities results in a proportional change in the other quantity. For example, if
, then
is proportional to
by a factor of 2, or
is proportional to
by a factor of 1/2. In other words, if you change
by some fixed amount, then
changes by double that amount.
Mathematically this means there is some constant scaling factor
such that
. In question 4, we're told
varies directly with
, which means there's some
such that
![f(x)=kx^2](https://tex.z-dn.net/?f=f%28x%29%3Dkx%5E2)
We're given that
when
, so
![40=k\cdot2^2\implies40=4k\implies k=10](https://tex.z-dn.net/?f=40%3Dk%5Ccdot2%5E2%5Cimplies40%3D4k%5Cimplies%20k%3D10)
Then when
, we get
![f(5)=10\cdot5^2=10\cdot25=250](https://tex.z-dn.net/?f=f%285%29%3D10%5Ccdot5%5E2%3D10%5Ccdot25%3D250)
- - -
For
to "vary inversely" with one another means that a change in one quantity results in an "opposite" change in the other. In other words, if
increases, then
decreases, and vice versa.
Mathematically, this is the same as saying there is some fixed number
such that
.
For example, if
and we start with
, then
also. If we change to
, then
; if
, then
, and so on.
In question 5, we're told
varies inversely with
, so that there is some constant
for which
![x\,f(x)=k](https://tex.z-dn.net/?f=x%5C%2Cf%28x%29%3Dk)
When
, we have
:
![4\cdot6=k\implies k=24](https://tex.z-dn.net/?f=4%5Ccdot6%3Dk%5Cimplies%20k%3D24)
Then for
, we get
![8\,f(8)=24\implies f(8)=3](https://tex.z-dn.net/?f=8%5C%2Cf%288%29%3D24%5Cimplies%20f%288%29%3D3)