Acompany plans to sell a new type of vacuum cleaner for $280 each. the company’s financial planner estimates that the cost, y, of manufacturing the vacuum cleaners is a quadratic function with a y-intercept of 11,000 and a vertex of (500, 24,000). which system of equations can be used to determine how many vacuums must be sold for the company to make a profit?

Ok, so find where the intersect and find where selling price>cost

the price function we will p(x)
that is 280 each or p(x)=280x

the cost function is the quadratic one, we call it c(x)
it's a little trickier
luckily, we know the vertex
remember, vertex form
y=a(x-h)²+k
vertex is (h,k) and a is some constant
given that the vertex is (500,24000)
y=a(x-500)²+24000
find a
given the y intercept is 11000
a point on the quadatic function is (0,11000)
subsitut to find a

11000=a(0-500)²+24000
11000=a(-500)²+24000
11000=250000a+24000
minus 24000 from both sides
-13000=250000a
divide both sides by 250000

so

so we have
p(x)=280x and

find where they intersect
p(x)=c(x)

expanding we get


solving we get
x≈-4432.34 or 47.72
we can't sell -4432 vacums

they intersect at about 47 or 48 vaccums

find which one on top where
p(48)=13440
c(48)=13376.2

p(x)>c(x) when x>47.7
so round up to 48 because 47 will be below

the number of vaccums is 47
the equations are
and
p(x)=280x