QuestionCharge \ $9 per guest, average 39 guests. Find demand$ q(p) $, revenue$ R(p) $, profit$ P(p)$, and break-even fees.

Studdy Solution

STEP 1

Assumptions1. The cover charge is $9 per guest. The average number of guests per night is393. The relationship between the cover charge and the number of guests is linear4. The nightly revenue is the product of the cover charge and the number of guests5. The club's nightly costs are given by the function$ C(p)=-25.5p+488$
6. The profit is the difference between the revenue and the costs

STEP 2

We first need to find the linear demand equation showing the number of guests $q$ per night as a function of the cover charge $p$ . We know that when the cover charge is $9, the number of guests is39. We can use this information to find the slope of the demand equation.
The slope of the demand equation is given by the change in $q$ divided by the change in $p$ . Since we only have one point, we can't calculate the slope directly. However, we can assume that the relationship is linear and that the number of guests decreases by1 for every $1 increase in the cover charge. This gives us a slope of -1.
$slope = -1$

STEP 3

Now that we have the slope, we can use the point-slope form of a line to find the demand equation. The point-slope form of a line is given by $q - q1 = slope \cdot (p - p1)$ where $(p1, q1)$ is a point on the line. We know that $(9,39)$ is a point on the line, so we can plug these values into the equation.

STEP 4

Substitute the values of $slope$ , $p1$ and $q1$ into the point-slope form of the line.
$q -39 = -1 \cdot (p -9)$

STEP 5

implify the equation to get the demand equation in slope-intercept form ( $q = mp + b$ ), where $m$ is the slope and $b$ is the y-intercept.
$q(p) = -p +48$

STEP 6

Next, we need to find the nightly revenue $R$ as a function of the cover charge $p$ . The revenue is the product of the cover charge and the number of guests, so we can express the revenue as $R(p) = p \cdot q(p)$

STEP 7

Substitute the demand equation $q(p)$ into the revenue function.
$R(p) = p \cdot (-p +48)$

STEP 8

implify the revenue function.
$R(p) = -p^2 +48p$

STEP 9

The profit $(p)$ is the difference between the revenue and the costs. We can express this as $(p) = R(p) - C(p)$

STEP 10

Substitute the revenue function $R(p)$ and the cost function $C(p)$ into the profit function.
$(p) = (-p^2 +48p) - (-25.5p +488)$

STEP 11

implify the profit function.
$(p) = -p^ +73.5p -488$

STEP 12

To find the entrance fees that allow Swing Haven to break even, we need to set the profit equal to zero and solve for $p$ .
$-p^2 +73.5p -488 =0$

STEP 13

This is a quadratic equation, and we can solve it using the quadratic formula $p = \frac{-b \pm \sqrt{b^2 -ac}}{2a}$ where $a = -$ , $b =73.5$ , and $c = -488$ .

STEP 14

Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula and solve for $p$ .
$p = \frac{-73. \pm \sqrt{(73.)^2 -4(-)(-488)}}{2(-)}$

STEP 15

implify the equation to find the values of $p$ .
$p = \frac{-73.5 \pm \sqrt{5402.25 -1952}}{-2}$

STEP 16

Further simplify the equation to find the values of $p$ .
$p = \frac{-73.5 \pm \sqrt{3450.25}}{-2}$

STEP 17

Calculate the square root and simplify the equation to find the values of $p$ .
$p = \frac{-73.5 \pm58.74}{-2}$

STEP 18

Calculate the two possible values of $p$ .
$p =7.38 \text{ or } p =66.12$ When the entrance fee is $7.38 or$ 66.12 per guest, then Swing Haven breaks even.

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