To say two quantities [tex]a[/tex] and [tex]b[/tex] "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 [tex]a=2b[/tex], then [tex]b[/tex] is proportional to [tex]a[/tex] by a factor of 2, or [tex]a[/tex] is proportional to [tex]b[/tex] by a factor of 1/2. In other words, if you change [tex]b[/tex] by some fixed amount, then [tex]a[/tex] changes by double that amount.
Mathematically this means there is some constant scaling factor [tex]k[/tex] such that [tex]a=kb[/tex]. In question 4, we're told [tex]f(x)[/tex] varies directly with [tex]x^2[/tex], which means there's some [tex]k[/tex] such that
[tex]f(x)=kx^2[/tex]
We're given that [tex]f(x)=40[/tex] when [tex]x=2[/tex], so
[tex]40=k\cdot2^2\implies40=4k\implies k=10[/tex]
Then when [tex]x=5[/tex], we get
[tex]f(5)=10\cdot5^2=10\cdot25=250[/tex]
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For [tex]a,b[/tex] 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 [tex]a[/tex] increases, then [tex]b[/tex] decreases, and vice versa.
Mathematically, this is the same as saying there is some fixed number [tex]k[/tex] such that [tex]ab=k[/tex].
For example, if [tex]k=1[/tex] and we start with [tex]a=1[/tex], then [tex]b=1[/tex] also. If we change to [tex]a=2[/tex], then [tex]b=\dfrac12[/tex]; if [tex]a=10[/tex], then [tex]b=\dfrac1{10}[/tex], and so on.
In question 5, we're told [tex]f(x)[/tex] varies inversely with [tex]x[/tex], so that there is some constant [tex]k[/tex] for which
[tex]x\,f(x)=k[/tex]
When [tex]x=4[/tex], we have [tex]f(x)=6[/tex]:
[tex]4\cdot6=k\implies k=24[/tex]
Then for [tex]x=8[/tex], we get
[tex]8\,f(8)=24\implies f(8)=3[/tex]
