A marketing analytics team is working on adjusting pricing for their popular new

Question

A marketing analytics team is working on adjusting pricing for their popular new smart phone. The cost of their phone is $4 per unit Currently the phone retails for $299 and sells about 2000 units per month. In a test market they determine that increasing the price by $10 will sales by about 50 units across their entire market. Write a linear model n(x) where n is the number of units sold in a month and x is the price of the phone. (Simplify and write in slope-intercept form.) Write another linear model p(x) where p is the profit per phone and x is the price of the phone Write a function T(x) where T is the total profit in a given month and x is the price of the phone. Determine the price that will earn the maximum profit in a month and the profit that this price will earn.

Explanation / Answer

Cost funrtcion C(x) = 4*x

Retail price (n(x) , x) : (2000, $ 299) and ( 1950 , $309)

slope = ( 2000 - 1950)/( 299 - 309) = 50/-10 = -5

n(x) = -5x + c; 2000 = -5*299 + c ; c= 3495

a) n(x) = -5x + 3495

b) profit = selling price - cost price = -5x + 3495 - 4x = -9x + 3495

p(x) = -9x + 3495

c) Total profit , T(x) = x(-9x+3495) = -9x^2 + 3495x

d) Maximum rpofit would ocuur at vertex of quadratic:

x = - (3495/2*-9) = $194.17

Maximum profit : T(194.17) = $ 339306.25

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