Question

I don’t understand this.. will someone please help? Will mark Brainliest along with 20 points.

Answer 1

To say two quantities $a$ and $b$ "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 $a=2b$, then $b$ is proportional to $a$ by a factor of 2, or $a$ is proportional to $b$ by a factor of 1/2. In other words, if you change $b$ by some fixed amount, then $a$ changes by double that amount.

Mathematically this means there is some constant scaling factor $k$ such that $a=kb$. In question 4, we're told $f(x)$ varies directly with $x^2$, which means there's some $k$ such that

$f(x)=kx^2$

We're given that $f(x)=40$ when $x=2$, so

$40=k\cdot2^2\implies40=4k\implies k=10$

Then when $x=5$, we get

$f(5)=10\cdot5^2=10\cdot25=250$

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For $a,b$ 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 $a$ increases, then $b$ decreases, and vice versa.

Mathematically, this is the same as saying there is some fixed number $k$ such that $ab=k$.

For example, if $k=1$ and we start with $a=1$, then $b=1$ also. If we change to $a=2$, then $b=\dfrac12$; if $a=10$, then $b=\dfrac1{10}$, and so on.

In question 5, we're told $f(x)$ varies inversely with $x$, so that there is some constant $k$ for which

$x\,f(x)=k$

When $x=4$, we have $f(x)=6$:

$4\cdot6=k\implies k=24$

Then for $x=8$, we get

$8\,f(8)=24\implies f(8)=3$