Math

QuestionNext week, you charged \$9 per guest with an average of 39 guests.
(a) Find the demand equation q(p)=q(p)=. (b) Find revenue R(p)=R(p)=. (c) Given C(p)=25.5p+488C(p)=-25.5p+488, find profit P(p)=P(p)=. (d) Determine break-even entrance fees, rounded to two decimal places.

Studdy Solution

STEP 1

Assumptions1. The cover charge per guest is 9.Theaveragenumberofguestspernightis393.Thedemandequationislinear4.Thenightlyrevenueistheproductofthecoverchargeandthenumberofguests5.Theclubsnightlycostsaregivenbythefunction9. The average number of guests per night is393. The demand equation 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)=-25.5 p+488$
6. The profit is the difference between the nightly revenue and the nightly costs

STEP 2

We start by finding a linear demand equation. A linear equation has the form y=mx+by = mx + b, where mm is the slope and bb is the y-intercept. In this case, the slope is the change in the number of guests per change in the cover charge, and the y-intercept is the number of guests when the cover charge is $0.
Since we only have one point (p=9p=9, q=39q=39), we can't directly calculate the slope. However, we can assume that the number of guests decreases linearly with the increase in cover charge. This means that the slope is negative. We'll denote the slope as k-k, where kk is a positive constant. So, the demand equation isq(p)=kp+bq(p) = -k p + b

STEP 3

To find the y-intercept bb, we can substitute the given point into the equation39=k9+b39 = -k \cdot9 + b

STEP 4

We don't know the value of kk, but we can express bb in terms of kkb=39+9kb =39 +9kSo, the demand equation isq(p)=kp+39+9kq(p) = -k p +39 +9k

STEP 5

Next, we find the nightly revenue RR as a function of the cover charge pp. The revenue is the product of the cover charge and the number of guests, soR(p)=pq(p)R(p) = p \cdot q(p)

STEP 6

Substitute the demand equation into the revenue functionR(p)=p(kp+39+9k)R(p) = p \cdot (-k p +39 +9k)

STEP 7

Expand the right side of the equationR(p)=kp2+39p+9kpR(p) = -k p^2 +39p +9kp

STEP 8

Combine like termsR(p)=kp2+(39+k)pR(p) = -k p^2 + (39 +k)p

STEP 9

Next, we find the profit in terms of the cover charge $p$. The profit is the difference between the revenue and the costs, so(p) = R(p) - C(p)$$

STEP 10

Substitute the revenue and cost functions into the profit function(p)=(kp2+(39+9k)p)(25.5p+488)(p) = (-k p^2 + (39 +9k)p) - (-25.5 p +488)

STEP 11

implify the right side of the equation(p)=kp+(39+9k+25.5)p488(p) = -k p^ + (39 +9k +25.5)p -488

STEP 12

To find the cover charges that allow the club to break even, we set the profit equal to 00 and solve for pp0=kp2+(39+9k+25.5)p4880 = -k p^2 + (39 +9k +25.5)p -488This is a quadratic equation in the form 0=ax2+bx+c0 = ax^2 + bx + c, and its solutions are given by the quadratic formulap=b±b24ac2ap = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}

STEP 13

In this case, a=ka = -k, b=39+9k+25.5b =39 +9k +25.5, and c=488c = -488. Substitute these values into the quadratic formula to find the values of pp.
Note Since we don't know the value of kk, we can't calculate the exact values of pp. However, we can express pp in terms of kk.

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