1.You have just opened a new nightclub, Russ\' Techno Pitstop, but are unsure of

Question

1.You have just opened a new nightclub, Russ' Techno Pitstop, but are unsure of how high to set the cover charge (entrance fee). One week you charged $5 per guest and averaged 300 guests per night. The next week you charged $8 per guest and averaged 240 guests per night.

(a) Find a linear demand equation showing the number of guests q per night as a function of the cover charge p.

(b) Find the nightly revenue R as a function of the cover charge p.

(c) The club will provide two free non-alcoholic drinks for each guest, costing the club $2 per head. In addition, the nightly overheads (rent, salaries, dancers, DJ, etc.) amount to $1,000. Find the cost C as a function of the cover charge p.

(d) Now find the profit in terms of the cover charge p.

Determine the entrance fee you should charge for a maximum profit.
p = $  per guest

2. As Sales Manager for Montevideo Productions, Inc., you are planning to review the prices you charge clients for television advertisement development. You currently charge each client an hourly development fee of $2,900. With this pricing structure, the demand, measured by the number of contracts Montevideo signs per month, is 11 contracts. This is down 3 contracts from the figure last year, when your company charged only $2,600.

(a) Construct a linear demand equation giving the number of contracts q as a function of the hourly fee p Montevideo charges for development.

(b) On average, Montevideo bills for 50 hours of production time on each contract. Give a formula for the total revenue obtained by charging $p per hour.

(c) The costs to Montevideo Productions are estimated as follows:

Express Montevideo Productions' monthly cost as a function of the number q of contracts.

Express Montevideo Productions' monthly cost as a function of the hourly production charge p.

(d) Express Montevideo Productions' monthly profit as a function of the hourly development fee p.

Find the price it should charge to maximize the profit.
p = $  per hour

Explanation / Answer

SOLUTION:

(1)

(a) We can do this by assuming a straight line equation , y= mx + c

Now , so clearly it gives us two equations from the given problem and that are:

300 = 5m +c .............................................................................(1)

240 = 8m +c ..............................................................................(2)

Now on slving equation (1) and (2), we get ,

m =- 20 and y =400

So the equation is y=400 - 20x

In terms of q(p), we have

q(p) = 400 - 20p

(b) Nightly revenue R as a function of a cover change p

So revenue [R]=qp = (400 -20p)*p =400p - 20p2

(c) Next cost [C]= 1000 + 2*q= 1000+2*(400- 20p)

(d) Now profit = revenue - cost

= (400p - 20p2 ) - [1000+2*(400- 20p) ]

= 400p - 20p2 -1800 + 40p

= 440p  - 20p2 -1800

This is a parabola opening down. In this you can easily figure out the maximum,

This is the plot that i have made

http://www.wolframalpha.com/input/?i=plot+440*p+-+20*p%C2%B2+-+1000+from+p+%3D+10+to+p+%3D+12

So we can clearly see that the maximum is at 11 , so p =11 dollars

(2)

a) Clearly we have two points: ($2,900, 11 contracts per month) and ($2,600, 14 contracts per month).

Now we will find the slope of the line:

m = (11 contracts per month - 14 contracts per month)/($2,900 - $2,600) = -3 contracts per month/$300 = -1 contract per month/$100

which means that every $100 we raise the price, we lose a contract.

Then use either point to find an equation:

(q - 11 contracts per month) = -1 contract per month/$100 (p - $2,600)
q - 11 contracts per month = -1 contract per month/$100 × p + 26 contracts per month
q(p) = -1 contract per month/$100 × p + 37 contracts per month

oer we can say q(p) = -0.01p + 37

b) This is 50 hours per contract times $p per hour times the total number of contracts from part a:

(-1 contract per month/$100 × p + 37 contracts per month) × 50 hours/contract per month × $p/hour =
TR(p) = -$(1/2 × p²) + $(1,850 × p)

or we can say TR(p) = -1/2 p² + 1,850 p

c) Total cost = fixed cost + variable cost per contract × number of contracts:

i) TC(q) = $140,000 per month + $80,000/contract × q contracts per month

or TC(q) = 140,000 + 80,000q

ii) As a function of p, we can use the formula above for q in terms of p:

TC(p) = $140,000 per month + $80,000/contract × (-1 contract per month/$100 × p + 37 contracts per month)

or

TC(p) = $140,000 per month - $(800 × p) per month + $2,960,000 per month

or

(TC(p) = 3,100,000 - 800p)

(d):

TR(p) = -1/2 p² + 1,850p

TC(p) = 3,100,000 - 800p

And since total profit per month = total revenue per month - total cost per month:

TP(p) = TR(p) - TC(p)
TP(p) = -1/2 p² + 1,850p - (3,100,000 - 800p)
TP(p) = -1/2 p² + 2,650p - 3,100,000

For finding the best price we will go with the derivative method:

Derivative method (calculus):

Take the derivative and set it equal to zero:

TP'(p) = -p + 2650 = 0, so p = $2,650 per contract will maximize the monthly profit.  

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