Which Catering Service Maximizes Charity Profit for a Business Banquet?

[Anj Estrada]

/files/483341/math134_project1.docx

/files/483341/Technical_Worksheet_for_project1.docx

If the links doesn't work, here is a copy.

Project on Math 134 Business Calculus, Summer 2015-2016
Project Due: 07 June 2016

Overview:

The company you are working for has decided to put on a banquet to raise money for a charity and must decide between two catering services to supply the food and servers. You have been assigned to the Banquet Planning Committee. You should work in a team with up to 6 members, each bringing their own individual strengths to the group. You will submit your work, neatly hand-written and clearly labelled. You must use algebra to solve the problems (not graphing calculator features). To best maintain accuracy, round only your final answers. Simplify completely.

Research Provided:

The Company’s research department has provided the following estimates:

• A demand of 230 banquet attendees can be expected at a dinner plate price of \$80.00 each. A demand of 370 banquet attendees can be expected at a dinner plate price of \$45.00 each.
• Catering Service A has a fixed cost of \$1,900 and a marginal cost of \$30 for each plate.
• Catering Service B has a fixed cost of \$3,000 and a marginal cost of \$22 for each plate.
• Costs for both caterers include the food, drinks, plates, utensils, tablecloths, glasses, crew, and cleanup.
• Dinner plates will only be sold as an entire unit. To justify company resources and to ensure the event will benefit the charity, the CEO insists the tickets be sold for no less than \$40. All profits will go toward a charity of the committee's choosing.
• Additional spontaneous donation to the charity will be accepted the night of the banquet. Studies estimate that 5% will give \$5, 23% will give \$20, 18% will give \$50, 7% will give \$100, and 2% will give \$500.

Analysis:

Each team should perform the following analyses:

1. Assume the price-demand function is linear. Use the research estimates to find the relationship between the price p, and the number of banquet attendees demanded, x. Find the relevant domain.

2. Find the revenue function, R(x), in terms of the number of banquet attendees x. Find the relevant domain by considering realistic limitations on the number of attendees and on price. Provide a sketch of graph.

3. Assume the cost function is linear and use the research estimates to find the cost function for each of the two possible catering services in terms of the number of banquet attendees x. Provide a sketch of graph.

4. Determine the break-even quantities for each of the two possible catering services.

5. Find the Profit function, P(x), for each of the two possible catering services in terms of the number of banquet attendees x.

6. If it is projected that there will be 100 tickets sold at a dinner price of \$112.50, which catering service should the committee recommend in order to earn the most profit for the charity?

7. Decide which catering service your company should choose if the projections yield 200 attendees. Include the ticket price at this demand.

8. Find the average cost function for Catering Service A. Evaluate the average cost per attendee if 50 tickets are purchased. Evaluate the average cost per attendee if 400 tickets are purchased.

9. On average, how much can you expect to receive in spontaneous donations for each banquet attendee? (Determine the per person expected value (weighted average) of donations.)

10. Overall recommendations: Which catering service does your committee recommend in order to obtain the most profit for the charity? How many people should attend the banquet to earn that profit and what profit can be expected from the banquet ticket sales? What price would you recommend for each dinner plate (ticket)? What dollar amount is expected from spontaneous donations? What will be the expected total amount raised for the charity (including the dinner ticket profits and spontaneous donations)?

Here is the technical worksheet:

Technical Worksheet

Group __________________

1. The linear price-demand function is p = D(x) = __________________________
Domain: _________________________________

2. The revenue function is R(x) = ______________________________
Domain: _________________________________

3. The cost functions for each of the two possible catering services in terms of x are

CA (x) = _______________________________________ and
CB (x) = _______________________________________.​

4. Catering Service A will at least break even at a minimum of ________ banquet attendees and a maximum of _______ banquet attendees.
Catering Service B will at least break even at a minimum of ________ banquet attendees and a maximum of _______ banquet attendees.

5. The profit functions for each of the possible Contractors in terms of x are

PA (x) = ____________________________________________ and
PB (x) = ____________________________________________.​

6. If the projected sales are 100 banquet tickets at a unit price of \$112.50, your company should choose Catering Service ______ in order to maximize the banquet’s profit since the profit using Catering Service A is \$_____________ and the profit using Catering Service B is \$____________.

7. If the projected sales are 200 banquet tickets at a unit price of \$__________, your company should choose Catering Service ______ in order to maximize the banquet’s profit since the profit using Catering Service A is \$____________ and the profit using Catering Service B is \$____________.

8. The average cost function for Catering Service A is CA(x)=_______________________________.

a. The average cost per attendee if 50 tickets are purchased is CA(50) \$______________.
b. The average cost per attendee if 400 tickets are purchased is CA(400) \$______________.​

9. On average, the expected value of spontaneous donations is \$_____________ per attendee.

10. The Committee would like to recommend Catering Service _________.
Number of Attendees for Maximum Profit: ______________
Maximum Charity Ticket Profit: \$______________
Optimal Number of Attendees: ______________
Optimal Charity Ticket Profit: \$______________
Banquet Ticket Price for Maximum Profit: \$______________
Optimal Banquet Ticket Price: \$______________
Maximum Expected Spontaneous Donations: \$______________
Optimal Spontaneous Donations: \$______________

Hope the problem is clear and I hope you can help me! Thank you very much :)

 

[MarkFL]

Hello and welcome to MHB, Anj Estrada! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Without knowing what you may have already done, let's begin with the first part of the analysis, which is to relate price $p$ and demand $x$. We are told to assume this relationship is linear, and we are given 2 points on the line:

\(\displaystyle (230,80),\,(370,45)\)

First, we need to determine the slope $m$ of the line, given by:

\(\displaystyle m=\frac{\Delta p}{\Delta x}=\frac{p_2-p_1}{x_2-x_1}\)

What do you get for the slope of the line?

 

[MarkFL]

After waiting several days for some feedback from the OP, I'm going to go ahead and now provide the answers for the benefit of the MHB community.

1.) As I stated, we need to find the slope of the price-demand line:

\(\displaystyle m=\frac{45-80}{370-230}=-\frac{1}{4}\)

Now we have the slope, and we have two given points, so we can use the point-slope formula to determine our line. I will use the first point:

\(\displaystyle p=-\frac{1}{4}(x-230)+80=-\frac{1}{4}x+\frac{275}{2}\)

Now, since we can't have a negative number of attendees, and the minimum price has been set at \$40, I would say our relevant domain is:

\(\displaystyle [0,390]\)

2.) Revenue $R$ is price per attendee times the number of attendees, hence:

\(\displaystyle R(x)=x\cdot p(x)=x\left(-\frac{1}{4}x+\frac{275}{2}\right)=\frac{x}{4}(550-x)\)

We see that the revenue function has roots of $x\in\{0,550\}$, and is a downward opening parabola. It's axis of symmetry will be midway between the roots, or at $x=275$, and so the vertex is at:

\(\displaystyle \left(275,\frac{75625}{4}\right)\)

The relevant domain is naturally the same as that which we found for the price function.

3.) Using the given information, we find the linear cost functions to be:

\(\displaystyle C_A(x)=30x+1900\)

\(\displaystyle C_B(x)=22x+3000\)

4.) The break-even point is the number of attendees that result in revenue and cost being equal, or where profit $P$ is zero, and profit is revenue minus costs. So, let's first determine the profit functions for the two catering companies:

\(\displaystyle P_A(x)=R(x)-C_A(x)=\frac{x}{4}(550-x)-(30x+1900)=\frac{1}{4}\left(-x^2+430x-7600\right)\)

\(\displaystyle P_B(x)=R(x)-C_B(x)=\frac{x}{4}(550-x)-(22x+3000)=\frac{1}{4}\left(-x^2+462x-12000\right)\)

Using the quadratic formula and rounding to the nearest integers for $x$ (the number of attendees is discrete rather than continuous), we find company $A$ has break even points at:

\(\displaystyle x\in\{18,412\}\)

And company $B$ has break even points at:

\(\displaystyle x\in\{28,434\}\)

5.) We have already determined the profit functions for both catering companies.

6.) We need to compute:

\(\displaystyle P_A(100)=\frac{1}{4}\left(-100^2+430(100)-7600\right)=6350\)

\(\displaystyle P_B(100)=\frac{1}{4}\left(-100^2+462(100)-12000\right)=6050\)

We see then that with 100 attendees, catering company $A$ will return the larger profit.

7.) With 200 attendees, the ticket price will be:

\(\displaystyle p(200)=-\frac{1}{4}200+\frac{275}{2}=87.50\)

The profit functions are as follows:

\(\displaystyle P_A(200)=\frac{1}{4}\left(-200^2+430(200)-7600\right)=9600\)

\(\displaystyle P_B(200)=\frac{1}{4}\left(-200^2+462(200)-12000\right)=10100\)

So, at this attendance level we see that catering company $B$ returns the larger profit.

8.) The average cost $\overline{C}$ is the total cost divided by the number of attendees:

\(\displaystyle \overline{C_A}(x)=\frac{30x+1900}{x}=30+\frac{1900}{x}\)

\(\displaystyle \overline{C_A}(50)=30+\frac{1900}{50}=68\)

\(\displaystyle \overline{C_A}(400)=30+\frac{1900}{400}=34.75\)

9.) The weighted average for donations is:

\(\displaystyle \overline{D}=(0.45)0+(0.05)5+(0.23)20+(0.18)50+(0.07)100+(0.02)500=30.85\)

So, the total expected donations is then:

\(\displaystyle D(x)=30.85x\)

10.) In order to determine which catering company to recommend, we need to look at the maximum of the two profit functions, found by setting the marginal profit to zero.

For catering company $A$

\(\displaystyle P_A'(x)=\frac{1}{4}\left(-2x+430\right)=0\implies x=215\)

At this attendance level, the price per ticket, the profit from company $A$ and the expected donations is:

\(\displaystyle p(215)=83.75\)

\(\displaystyle P_A(215)=\frac{1}{4}\left(-215^2+430(215)-7600\right)=9656.25\)

\(\displaystyle D(215)=6632.75\)

Going with company $A$, then we can expect a total of \$16289 to be raised for the charity.

\(\displaystyle P_B'(x)=\frac{1}{4}\left(-2x+462\right)=0\implies x=231\)

At this attendance level, the price per ticket, the profit from company $A$ and the expected donations is:

\(\displaystyle p(231)=79.75\)

\(\displaystyle P_B(231)=\frac{1}{4}\left(-231^2+462(231)-12000\right)=10340.25\)

\(\displaystyle D(231)=7126.35\)

Going with company $B$, then we can expect a total of \$ 17466.60 to be raised for the charity.

Clearly, we should recommend that catering company $B$ be used for the banquet.