Question

As Sales Manager for ISeeYou Productions, you are planning to review the prices you charge clients for television advertisement development. You currently charge each client a development fee of $7,200. With this pricing structure, ISeeYou is able to sign 6 contracts per month. This is down from 14 contracts, which was the figure last year when your company charged each client only $4,000. (a) Construct a linear function that yields the development fee p that ISeeYou should charge in order to sign q contracts per month (B) Find the total revenue R ISeeYou obtains by signing q contracts.(c) The costs to ISeeYou Productions are estimated as follows:
Fixed costs: $16,000 per month
Variable costs: 800q dollars (when q contracts are signed)
Express ISeeYou Productions' monthly cost as a function of the number q of contracts.
C(q) =


(d) Express ISeeYou Productions' monthly profit as a function of the number q of contracts.
P(q) = .

(e) How many contracts could ISeeYou sign to break even? (Enter the lower value first.)
ISeeYou breaks even when they sign

Answer 1

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: $R(q)=-400q^{2}+9600q$. 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: $P(q)=-400p^{2}+8800q-16000$. 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:

$m=\frac{p_{2}-p_{1}}{q_{2}-q_{1}}$

$m=\frac{4000-7200}{14-6}$

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:

$y-y_{1}=m(x-x_{1})$

$p-p_{1}=m(q-q_{1})$

if we used the point: (6, 7200) we get:

$p-7200=-400(q-6)$

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

$R(q)=(-400q+9600)q$

$R(q)=-400q^{2}+9600q$

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

$P(q)=-400q^{2}+9600q-(800q+16000)$

so we can now simplify our function to get:

$P(q)=-400q^{2}+9600q-800q-16000$

$P(q)=-400q^{2}+8800q-16000$

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:

$P(q)=-400q^{2}+8800q-16000$

$-400q^{2}+8800q-16000=0$

we can start by factoring a -400 from the left side so we get:

$-400(q^{2}-22q+40)=0$

we can divide both sides of the equation into -400 to simplify the equation so we get:

$q^{2}-22q+40=0$

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.

brainly.com/question/23407629?referrer=searchResults