Question

20 points! Can someone do part A for me? I know how to do these questions I’m just a tad bit confused. :/

Answer 1

Revenue = (number of phones)(price of phones)

If Mobistar charges $80 per phone, then their revenue is

$r=800\times80=64,000$

If they instead charge $2 less per phone (so that $d=1$), or $78 per phone, then their revenue would be

$r=840\times78=65,520$

If they charge $4 less per phone ($d=2$), then they make

$r=880\times76=66,880$

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 $d$; 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 $d$, since the cost of a phone starts at a fixed $80 and decreases by $2 for each unit of $d$. So total revenue can be written as

$r(d)=(800+40d)(80-2d)$

You can stop there, but you may be expected to write this in standard quadratic form, which is just a matter of expanding $r(d)$.

$r(d)=64,000+1,600d-80d^2$

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

$r(d)=-80d^2+1,600d+64,000$
$r(d)=-80(d^2-20d-800)$
$r(d)=-80(d^2-20d+100-100-800)$
$r(d)=-80((d-10)^2-900)$
$r(d)=72,000-80(d-10)^2$

which is to say the vertex of the parabola described by $r(d)$ 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 $d=10$, netting them $72,000. This translates to setting the price per phone to $80-2(10)=\ 60$.