Question

You have just opened a new dance club, Swing Haven, but are unsure of how high to set the cover charge (entrance fee). One week you charged $7 per guest and averaged 79 guests per night. The next week you charged $16 per guest and averaged 43 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.q(p) = (b) Find the nightly revenue R as a function of the cover charge p.R(p) = (c) When you set the admission to p dollars, the club's nightly costs, including rent, salaries, and two free non-alcoholic drinks for each guest, amounts toC(p) =−26.75p +939Find the profit in terms of the cover charge p.P(p) = (d) Determine entrance fees that allow Swing Haven to break even. Enter the lower fee first, and round your answer to two decimal places.When the entrance fee is p = or dollars per guest, then Swing Haven breaks even.

Answer 1

Answer:

a) The demand function is

$q(p) = -4 p + 107$

b) The nightly revenue is

$R(p) = -4 p^2 + 107 p$

c) The profit function is

$P(p) = -4 p^2 + 133.75 p - 939$

d) The entrance fees that allow Swing Haven to break even are between 10.03 and 23.41 dollars per guest.

Step-by-step explanation:

a) Lets find the slope s of the demand:

$s = \frac{79-43}{7-16} = \frac{36}{-9} = -4$

Since the demand takes the value 79 in 7, then

$q(p) = -4 (p-7) + 79 = -4 p + 107$

b) The nightly revenue can be found by multiplying q by p

$R(p) = p*q(p) = p*( -4 p + 107) = -4 p^2 + 107 p$

c) The profit function is obtained from substracting the const function C(p) from the revenue function R(p)

$P(p) = R(p) - C(p) = p*q(p) = -4 p^2 + 107 p - (-26.75p + 939) = \\\\-4 p^2 + 133.75 p - 939$

d) Lets find out the zeros and positive interval of P. Since P is a quadratic function with negative main coefficient, then it should have a maximum at the vertex, and between the roots (if any), the function should be positive. Therefore, we just need to find the zeros of P

$r_1, r_2 = \frac{-133.75 \,^+_-\, \sqrt{133.75^2-4*(-4)*(-939)} }{-8} = \frac{-133.75 \,^+_-\, 53.526}{-8} \\r_1 = 10.03\\r_2 = 23.41$

Therefore, the entrance fees that allow Swing Haven to break even are between 10.03 and 23.41 dollars per guest.