1 One summer after taking calculus Dustin and Lauren start a

1. One summer after taking calculus Dustin and Lauren start a small business selling their painting services. They decide to illustrate their new found calculus abilities. To simplify things they work by the hour as a team and as a starting wage they charge $52/per hour. At this wage they seem to consistently book 40 hours of work each week. After a few weeks they start reducing their hourly wage in increments of $2 in attempt to see the effect on their Revenue. They discover that for every $2 price reduction, they get an additional 4 hours of work! Determine the price and quantity functions in terms of the number of $2 price reductions. Find the price that they should charge to maximize their Revenue!

Let x represent the number of $2 price reductions.

Solution

Ideal situation :
Wage = $52 per hr
No of hrs = 40

Every 2$ reduction in wage, results in extra 4 hrs

So, wage = 52 - 2x dollars
No of hrs = 40 + 4x

Revenue = (52 - 2x)(40 + 4x)

Multiply it out :

R = -8x^2 + 128x + 2080

Lets find the vertex of this :
R = -8(x^2 - 12x - 260)

R = -8(x^2- 12x + 36 - 36 - 260)

R = -8(x^2 - 12x + 36) - 8(-296)

R = -8(x - 6)^2 + 2368

So, 6

So, price charged is :
52 - 2x dollars
52 - 2*6
52 - 12

40 dollars per hr ----> ANSWER