Answer:
(i) The x -intercepts are [tex]x=20[/tex] and[tex]x=330[/tex].
(ii) The maximize profit occurs at [tex]$f(175)=-.5 \times(175)^{2}+175 \times(175)-3,300=\12,012.50$\\[/tex]
Step-by-step explanation:
Given :
[tex]p(x)=-0.5 x^{2}+175 x-3300[/tex]
Step 1
Solve p(x) by using quadratic equation,
[tex]-0.5 x^{2}+175 x-3300= 0[/tex]
Multiply both sides of the equation by 10,
[tex](10-0.5 x^{2}+175 x-3300)= 0*10[/tex]
[tex]$-0.5 x^{2} \cdot 10+175 x \cdot 10-3300 \cdot 10=0 \cdot 10$[/tex]
[tex]$-5 x^{2}+1750 x-33000=0$[/tex]
Step 2
Solve with the quadratic formula
[tex]$x=\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$[/tex]
Put [tex]a=-5,b=1750 and c=-33000[/tex]
[tex]$x_{1,2}=\frac{-1750 \pm \sqrt{1750^{2}-4(-5)(-33000)}}{2(-5)}$[/tex]
[tex]$\sqrt{1750^{2}-4(-5)(-33000)}=1550$[/tex]
[tex]$x_{1,2}=\frac{-1750 \pm 1550}{2(-5)}$[/tex]
Step 3
Separating the solutions, we get
[tex]$x_{1}=\frac{-1750+1550}{2(-5)}$[/tex]
[tex]x_{1} =20[/tex] and
[tex]$x_{2}=\frac{-1750-1550}{2(-5)}$[/tex]
[tex]x_{2} =330[/tex]
Step 4
(ii)The maximum units for maximum profit is realized at the point
[tex]$x=-\frac{b}{2 a}$[/tex]
[tex]$x=-\frac{175}{2 \cdot(-.5)}=175$[/tex]
Maximum profit occurs at
[tex]$f(175)=-.5 \times(175)^{2}+175 \times(175)-3,300[/tex]
[tex]= 12,012.50$[/tex]
The maximize profit occurs at [tex]$f(175)=\$ 12,012.50$[/tex]