help please please

Question

a technology company can sell a special app to businesses for $40 each. 260 businesses are willing to purchase the app at this price. for every $5 increase in price there are 10 fewer businesses who are willinh to buy the app. since revenue is a function of the number of app sold we use equation R=(40+5)(250-10) to represent the revenue of positive number for price change.

a) maximum revenue the company can expect

b) price should they charge for each app if they want to make the maximum amount of money

c) what is the y intercept represent in real life

d) positive x intercepts represent in real life
 

Answer 1

The price special app is $40 .

Number of businesses that are willing to purchase app at this price is 260

For every $5 increase in price of the app 10 businesses decreases who  are willing to buy .

If x times the price $40 increases by $5 then x times of 10 businesses  will decreases .

Mathematically the  revenue function R = Price of each app * number of businesses .

                                                           R = (40 + 5*x )( 260 - 10*x)

(a)

To find the maximum revenue , make the first derivative of revenue function to zero .

 R = (40 + 5*x )( 260 - 10*x)

 R = 10400 + 1300x - 400x - 50 x²

 R =   - 50 x² +900x +10400 

Find its first derivative .

 R' =   - 100x +900 =0

 - 100x +900 =0

 100x = 900 

x = 9

Now put x = 9 in revenue function to get maximum revenue .

R = (40 + 5*x )( 260 - 10*x)

R = (40 + 5*9)( 260 - 10*9)

R = (40 + 45 )( 260 - 90)

R = (85)( 170)

R = 14450 .

So the maximum revenue the company can expect is 14450 .

Answer 2

(b)

The price special app is $40 .

Number of businesses that are willing to purchase app at this price is 260

For every $5 increase in price of the app 10 businesses decreases who  are willing to buy .

If x times the price $40 increases by $5 then x times of 10 businesses  will decreases .

The obtained value of x is 9              [ From (a) ]

So the price of each app is = 40 + 5*x 

                                     = 40 + 5*9

                                     = 40 + 45

                                     = 85

The price of each app to get maximum revenue is 85$ 

Answer 3

(c)

The obtained revenue function is R =   - 50 x² +900x +10400 .

To find y-intercept put x = 0 in revenue function .

R = - 50 0² +900(0) +10400 .

R =10400 

Hence y-intercept represent the revenue of the company .

The  y-intercept is R =10400.