revenue = (number of phones)(price of phones)
If Mobistar charges $80 per phone, then their revenue is
[tex]r=800\times80=64,000[/tex]
If they instead charge $2 less per phone (so that [tex]d=1[/tex]), or $78 per phone, then their revenue would be
[tex]r=840\times78=65,520[/tex]
If they charge $4 less per phone ([tex]d=2[/tex]), then they make
[tex]r=880\times76=66,880[/tex]
and so on. As the hint suggests, we can write the number of phones sold as an expression depending on the number of price decreases [tex]d[/tex]; that is, for each price decrease, the number of phones sold increases by 40.
Similarly, the price per phone can be written in terms of [tex]d[/tex], since the cost of a phone starts at a fixed $80 and decreases by $2 for each unit of [tex]d[/tex]. So total revenue can be written as
[tex]r(d)=(800+40d)(80-2d)[/tex]
You can stop there, but you may be expected to write this in standard quadratic form, which is just a matter of expanding [tex]r(d)[/tex].
[tex]r(d)=64,000+1,600d-80d^2[/tex]
For the remaining parts:
(B) Writing in vertex form will help, as the name suggests. That's just a matter of completing the square. You have
[tex]r(d)=-80d^2+1,600d+64,000[/tex]
[tex]r(d)=-80(d^2-20d-800)[/tex]
[tex]r(d)=-80(d^2-20d+100-100-800)[/tex]
[tex]r(d)=-80((d-10)^2-900)[/tex]
[tex]r(d)=72,000-80(d-10)^2[/tex]
which is to say the vertex of the parabola described by [tex]r(d)[/tex] occurs at (10, 72,000). The parabola "opens downward" (which we know because the sign of the leading (quadratic) term is negative), so the vertex is a maximum of the revenue function, which in turn means the most revenue the company can achieve will be achieved if [tex]d=10[/tex], netting them $72,000. This translates to setting the price per phone to [tex]80-2(10)=\$60[/tex].
