The development fee p can be modeled by the equation p=-400q+9600 where p is the price the company charges and q is the number of contracts they get. The total revenue R ISeeYou obtains by signing q contracts is found by using the function: [tex]R(q)=-400q^{2}+9600q[/tex]. The cost ISeeYou Productions can be found by using the following function: C(q)=800q+16000. The monthlhy profit of Express ISeeYou Productions if found with the next function: [tex]P(q)=-400p^{2}+8800q-16000[/tex]. ISeeYou should sign 2 or 20 contracts to break even.
a) In order to solve part a of the problem, we will suppose the price will have a linear relation with the amount of contracts they signed, so we need to find two points to build this function.
We know that when signing 6 contracts, the company charges $7,200, so our first point will be:
(6, 7200)
we also know that the company signs 14 contracts when they charge $4,000. So our second point is:
(14, 4000)
so we can use these points to find the slope of the line by using the
slope formula:
[tex]m=\frac{p_{2}-p_{1}}{q_{2}-q_{1}}[/tex]
[tex]m=\frac{4000-7200}{14-6}[/tex]
so the slope is:
m=-400
We can now use one of those points and the slope to find the function for the price, so we get:
[tex]y-y_{1}=m(x-x_{1})[/tex]
[tex]p-p_{1}=m(q-q_{1})[/tex]
if we used the point: (6, 7200) we get:
[tex]p-7200=-400(q-6)[/tex]
which can now be simplified:
p=-400q+2400+7200
so the price function is:
p=-400q+9600
b) In order to find the revenue equation we need to multiply the price equation by the amount of contracts they signed in a month, so we get:
R(q)=pq
[tex]R(q)=(-400q+9600)q[/tex]
[tex]R(q)=-400q^{2}+9600q[/tex]
c) The cost can be found by adding the fixed cost and the variable cost, so we get:
C(q)=800q+16000
d) The profit is found by subtracting the montly cost from the monthly revenue, so we get:
P(q)=R(q)-C(q)
so
[tex]P(q)=-400q^{2}+9600q-(800q+16000)[/tex]
so we can now simplify our function to get:
[tex]P(q)=-400q^{2}+9600q-800q-16000[/tex]
[tex]P(q)=-400q^{2}+8800q-16000[/tex]
e) In order to find the number of contracts for the company to break even, we need to find the q contracts that will give a profit of $0 so we set the function of the previous part of the problem equal to zero so we get:
[tex]P(q)=-400q^{2}+8800q-16000[/tex]
[tex]-400q^{2}+8800q-16000=0[/tex]
we can start by factoring a -400 from the left side so we get:
[tex]-400(q^{2}-22q+40)=0[/tex]
we can divide both sides of the equation into -400 to simplify the equation so we get:
[tex]q^{2}-22q+40=0[/tex]
and now we can factor the left side of the equation to get:
(q-20)(q-2)=0
we can now split this into two equations to get:
q-2=0 and q-20=0
so our two answers are:
q=2 and q=20
the company will break even when signing 2 and 20 contracts.
In the attached picture you will be able to see the break even points on the graph of revenue and cost.
For more information on how to solve this type of problems, you can go to the next link.
https://brainly.com/question/23407629?referrer=searchResults
