QuestionNext week, you charge \$9 per guest with 39 guests on average.
(a) Find the demand equation $q(p)=$ . (b) Find revenue $R(p)=$ . (c) Costs $C(p)=-25.5p+488$ , find profit $P(p)=$ . (d) Determine break-even entrance fees $p=$ .

Studdy Solution

STEP 1

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

STEP 2

We will start by finding the linear demand equation. We know that when the cover charge is $9, the number of guests is39. We can represent this as a point on the graph$ (9,39) $.Since we are looking for a linear equation, we can assume that the equation is in the form$ q(p) = mp + b $, where$ m $is the slope and$ b$ is the y-intercept.

STEP 3

Since we only have one point, we can't find the slope. However, we can assume that the number of guests will be zero if the cover charge is too high. This gives us another point on the graph $(p,0)$ .We can find the slope using the formula $m = (y2 - y1) / (x2 - x1)$ .

STEP 4

Plug in the values for the points into the slope formula.
$m = (0 -39) / (p -9)$

STEP 5

olve for $p$ .
$p =9 -39/m$

STEP 6

Now we can write the demand equation as $q(p) = m(p -9) +39$

STEP 7

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

STEP 8

Substitute the demand equation into the revenue equation.
$R(p) = p \cdot [m(p -) +39]$

STEP 9

Next, we will find the profit as a function of the cover charge. The profit is the difference between the revenue and the costs.
$(p) = R(p) - C(p)$

STEP 10

Substitute the revenue and cost equations into the profit equation.
$(p) = [p \cdot m(p -9) +39p] - [-25.5p +488]$

STEP 11

Finally, we will find the cover charges that allow the club to break even. This occurs when the profit is zero.
Set the profit equation equal to zero and solve for $p$ .
$0 = [p \cdot m(p -9) +39p] - [-25.5p +488]$

STEP 12

olve the equation for $p$ to find the cover charges.Please note that the exact solution will depend on the value of $m$ , which we don't have. However, this is the general approach to solve the problem.

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QuestionNext week, you charge \$9 per guest with 39 guests on average.
(a) Find the demand equation $q(p)=$ . (b) Find revenue $R(p)=$ . (c) Costs $C(p)=-25.5p+488$ , find profit $P(p)=$ . (d) Determine break-even entrance fees $p=$ .

Studdy Solution

STEP 1

STEP 2

STEP 3

Since we only have one point, we can't find the slope. However, we can assume that the number of guests will be zero if the cover charge is too high. This gives us another point on the graph $(p,0)$ .We can find the slope using the formula $m = (y2 - y1) / (x2 - x1)$ .

STEP 4

Plug in the values for the points into the slope formula.
$m = (0 -39) / (p -9)$

STEP 5

olve for $p$ .
$p =9 -39/m$

STEP 6

Now we can write the demand equation as $q(p) = m(p -9) +39$

STEP 7

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

STEP 8

Substitute the demand equation into the revenue equation.
$R(p) = p \cdot [m(p -) +39]$

STEP 9

Next, we will find the profit as a function of the cover charge. The profit is the difference between the revenue and the costs.
$(p) = R(p) - C(p)$

STEP 10

Substitute the revenue and cost equations into the profit equation.
$(p) = [p \cdot m(p -9) +39p] - [-25.5p +488]$

STEP 11

Finally, we will find the cover charges that allow the club to break even. This occurs when the profit is zero.
Set the profit equation equal to zero and solve for $p$ .
$0 = [p \cdot m(p -9) +39p] - [-25.5p +488]$

STEP 12

olve the equation for $p$ to find the cover charges.Please note that the exact solution will depend on the value of $m$ , which we don't have. However, this is the general approach to solve the problem.

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QuestionNext week, you charge \$9 per guest with 39 guests on average. (a) Find the demand equation q(p)=q(p)=q(p)=. (b) Find revenue R(p)=R(p)=R(p)=. (c) Costs C(p)=−25.5p+488C(p)=-25.5p+488C(p)=−25.5p+488, find profit P(p)=P(p)=P(p)=. (d) Determine break-even entrance fees p=p=p=.

Studdy Solution

STEP 1

STEP 2

STEP 3

STEP 4

Plug in the values for the points into the slope formula.m=(0−39)/(p−9)m = (0 -39) / (p -9)m=(0−39)/(p−9)

STEP 5

olve for ppp.p=9−39/mp =9 -39/mp=9−39/m

STEP 6

Now we can write the demand equation asq(p)=m(p−9)+39q(p) = m(p -9) +39q(p)=m(p−9)+39

STEP 7

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

STEP 8

Substitute the demand equation into the revenue equation.R(p)=p⋅[m(p−)+39]R(p) = p \cdot [m(p -) +39]R(p)=p⋅[m(p−)+39]

STEP 9

Next, we will find the profit as a function of the cover charge. The profit is the difference between the revenue and the costs.(p)=R(p)−C(p)(p) = R(p) - C(p)(p)=R(p)−C(p)

STEP 10

Substitute the revenue and cost equations into the profit equation.(p)=[p⋅m(p−9)+39p]−[−25.5p+488](p) = [p \cdot m(p -9) +39p] - [-25.5p +488](p)=[p⋅m(p−9)+39p]−[−25.5p+488]

STEP 11

Finally, we will find the cover charges that allow the club to break even. This occurs when the profit is zero.Set the profit equation equal to zero and solve for ppp.0=[p⋅m(p−9)+39p]−[−25.5p+488]0 = [p \cdot m(p -9) +39p] - [-25.5p +488]0=[p⋅m(p−9)+39p]−[−25.5p+488]

STEP 12

olve the equation for ppp to find the cover charges.Please note that the exact solution will depend on the value of mmm, which we don't have. However, this is the general approach to solve the problem.

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionNext week, you charge \$9 per guest with 39 guests on average.
(a) Find the demand equation $q(p)=$ . (b) Find revenue $R(p)=$ . (c) Costs $C(p)=-25.5p+488$ , find profit $P(p)=$ . (d) Determine break-even entrance fees $p=$ .

Plug in the values for the points into the slope formula.
$m = (0 -39) / (p -9)$

olve for $p$ .
$p =9 -39/m$

Now we can write the demand equation as $q(p) = m(p -9) +39$

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

Substitute the demand equation into the revenue equation.
$R(p) = p \cdot [m(p -) +39]$

Next, we will find the profit as a function of the cover charge. The profit is the difference between the revenue and the costs.
$(p) = R(p) - C(p)$

Substitute the revenue and cost equations into the profit equation.
$(p) = [p \cdot m(p -9) +39p] - [-25.5p +488]$

Finally, we will find the cover charges that allow the club to break even. This occurs when the profit is zero.
Set the profit equation equal to zero and solve for $p$ .
$0 = [p \cdot m(p -9) +39p] - [-25.5p +488]$

olve the equation for $p$ to find the cover charges.Please note that the exact solution will depend on the value of $m$ , which we don't have. However, this is the general approach to solve the problem.