Table of Contents
Project Title
3
Problem Statement
4
Mathematical Formulation
5
Problem Solution
6
Discussion of Findings
10
Conclusion
11
Bibliography
12
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Project Title
An investigation to determine whether an investment in a boat tour business is feasible with a payback period of
3 years.
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Problem Statement
King Kong Tours located in Kingston, Jamaica is interested in adding a boat ride tour of the Kingston Harbour
to their offerings. The executive board will only invest the required 𝐽𝑀𝐷$100 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 in the new venture if the
payback period is less than 3 years.
Based on the available market data available, the monthly demand for boat ride tickets can be estimated by the
demand schedule:
Figure 1:
Monthly Demand Schedule for Boat Rides
Price ($ 000)
Number of Passengers
1
65
16
60
46
50
91
35
166
10
181
5
Additionally, the tour company incurs a fixed cost of 𝐽𝑀𝐷$14,000 per month to cover boat maintenance and pays
a fee of 𝐽𝑀𝐷$4,000 per passenger in insurance, taxes and other related fees.
This project aims to determine the whether the investment is feasible with an expected payback period of less
than 3 years. This will be achieved through the following objectives:
1.
2.
3.
4.
5.
Develop a model for the demand of boat rides offered by the company
Develop a model for the revenue generated from the boat rides offered by the company
Develop a model for the profit generated from the boat rides offered by the company
Determine the maximum possible profit from the boat rides.
Determine the expected payback period from the investment in the venture
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Mathematical Formulation
The problem at hand will be solved through the following steps:
Step 1:
Develop a linear model for the monthly demand of boat tours.
Step 2:
Develop a model for the monthly revenue from boat tours using
𝑅𝑒𝑣𝑒𝑛𝑢𝑒 = 𝑃𝑟𝑖𝑐𝑒 × 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦
Step 3:
Develop a model for the monthly cost for offering boat tours using
𝐶𝑜𝑠𝑡 = 𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡 + 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝐶𝑜𝑠𝑡
Step 4:
Step 5:
Develop a model for the monthly profit for the monthly profit from the offering of boat rides
using
𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑅𝑒𝑣𝑒𝑛𝑢𝑒 − 𝐶𝑜𝑠𝑡
Step 6:
Use the second derivative to confirm that the stationary point found is indeed a maximum point
Step 7:
Calculate the payback period for the investment using
𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
𝑃𝑎𝑦𝑏𝑎𝑐𝑘 𝑃𝑒𝑟𝑖𝑜𝑑 =
𝐴𝑛𝑛𝑢𝑎𝑙 𝑃𝑟𝑜𝑓𝑖𝑡
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Determine the maximum profit by finding the stationary point of the profit function
Problem Solution
Demand Model
Firstly, to develop the demand model, we must assume that the demand schedule is constant over time. This
assumption is necessary as the demand for leisure activities, such as boat rides, varies with the period of the
year. Therefore, it becomes necessary to assume a constant demand model for all months of the year.
Graphically, the demand model can be represented with a scatter plot as shown below in Figure 2.
Figure 2:
Scatter Plot of Price vs. Number of Tickets Sol
200
180
160
Price/ $000
140
120
100
80
60
40
20
0
0
10
20
30
40
50
60
70
Number of Tickets
Based on the scatter plot, we can observe that the demand schedule shows a strong negative correlation between
the price and number of tickets. This observation is in line with the law of demand which states that all things
being equal, as the price of a good increases, the quantity demanded of that good will decrease.
Since our points along the scatter form an apparent straight line, we can proceed with finding the demand model
in the form 𝑦 = 𝑚𝑥 + 𝑐. In doing so, the gradient can be calculated using the points (60,16) and (10,166).
Using the formula:
We get:
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𝑚=
𝑦 −𝑦
𝑥 −𝑥
𝑚=
166 − 16 150
=
= −3
10 − 60
−50
Using the point (35, 91) from our schedule, the demand model can be estimated using:
𝑝(𝑥) − 𝑦 = 𝑚(𝑥 − 𝑥 )
Where 𝑝(𝑥) is the price of the boat rides in thousands of dollars and 𝑥 is the number of passengers. Fitting this
model, we get:
𝑝(𝑥) − 91 = −3(𝑥 − 35)
𝑝(𝑥) = −3𝑥 + 105 + 91
That is:
𝑝(𝑥) = 196 − 3𝑥 , 𝑥 ≥ 0
Revenue Model
With this information, the revenue model can be estimated using
𝑅𝑒𝑣𝑒𝑛𝑢𝑒 = 𝑃𝑟𝑖𝑐𝑒 × 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦
Where the price comes from the demand function. Thus, we get:
𝑅(𝑥) = 𝑥 × 𝑝(𝑥)
𝑅(𝑥) = 𝑥(196 − 3𝑥)
That is:
𝑅(𝑥) = 196𝑥 − 3𝑥
Cost Model
The general cost model is defined as
𝑇𝑜𝑡𝑎𝑙 𝐶𝑜𝑠𝑡 = 𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡 + [𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝐶𝑜𝑠𝑡 𝑝𝑒𝑟 𝑈𝑛𝑖𝑡 × 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑈𝑛𝑖𝑡𝑠]
Based on the information provided, the fixed cost is $14,000 and the variable cost per passenger is $4,000.
Thus, the cost function is
𝐶(𝑥) = 14 + 4𝑥, 𝑥 ≥ 0
where 𝐶(𝑥) is the cost associated with providing the service to 𝑥 passengers per month.
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Profit Model
The general profit model is given by
𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑅𝑒𝑣𝑒𝑛𝑢𝑒 − 𝐶𝑜𝑠𝑡
Since we have obtained model for both revenue and cost, we can fit our model as:
𝑃(𝑥) = 196𝑥 − 3𝑥 − (14 + 4𝑥)
That is:
𝑃(𝑥) = 196𝑥 − 3𝑥 − 14 − 4𝑥
𝑃(𝑥) = −3𝑥 + 192𝑥 − 14, 𝑥 ≥ 0
Figure 3:
Graph of Profit Function
To determine the maximum possible profit, we will proceed with analyzing the marginal profit function.
The marginal profit function is:
𝑃 (𝑥) = −6𝑥 + 1 92
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The point where maximum profit is achieved is a stationary point of the profit function. Therefore, the
maximum profit occur when
𝑃 (𝑥) = 0
That is:
−6𝑥 + 192 = 0
6𝑥 = 192
𝑥 = 32
Therefore, the maximum profit occurs when 32 ride tickers are sold per month. The maximum profit can then
be calculated as:
𝑃
= 𝑃(32) = −3(32) + 192(32) − 14
𝑃
= 3058
Therefore, the maximum profit from the monthly sales of ride tickets is $3,058,000. Which implies that the
maximum profit for the year is 12 × $3,058,000 = $36,696,000.
To prove that this is indeed the maximum possible profit, we will analyze the sign of the second derivative at
the point 𝑥 = 32.
𝑃 (𝑥) = −6 < 0
Since the second derivative is less than zero, this confirms that our calculations have produced the maximum
possible profit for the boat ride venture under the given marker conditions.
Application of Solution
The payback period for an investment is defined by the formula
𝑃𝑎𝑦𝑏𝑎𝑐𝑘 𝑃𝑒𝑟𝑖𝑜𝑑 =
𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
𝐴𝑛𝑛𝑢𝑎𝑙 𝑃𝑟𝑜𝑓𝑖𝑡
Assuming the company can sell 32 ride tickets monthly to achieve maximum profit, the payback period can be
calculated as:
𝑃𝑎𝑦𝑏𝑎𝑐𝑘 𝑃𝑒𝑟𝑖𝑜𝑑 =
100,000,000
= 2.72 𝑦𝑒𝑎𝑟𝑠 (3 𝑠𝑓)
36,696,000
Since the payback period is less than the required of three years, the company should invest in the venture as
they would be able to recoup their investment from the profits generated before the required three years.
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Discussion of Findings
Based on the findings of this research, investing in a boat tour business with a 3-year payback period is feasible
for King Kong Tours. The analysis considered several key models: the demand model (𝑝(𝑥) = 196 − 3𝑥),
revenue model (𝑅(𝑥) = 196𝑥 − 3𝑥 ), cost model (𝐶(𝑥) = 14 + 4𝑥), and profit model (𝑃(𝑥) = −3𝑥 +
192𝑥 − 14).
The demand model indicates that as ticket prices increase, the number of tickets sold decreases. The revenue
model shows that revenue initially increases, reaches a maximum point, and then decreases, considering the
relationship between price and demand. The cost model reveals a linear increase in costs as the number of tickets
sold rises.
The profit model combines the revenue and cost models to determine profitability. It demonstrates that profit
increases, reaches a maximum point, and then decreases. It was found that the maximum profit occurs when 32
boat ride tickets are sold per month, resulting in a monthly profit of $3,058,000 and an annual profit of
$36,696,000.
The second derivative test confirms that the maximum profit is indeed a maximum point, adding credibility to the
findings. The payback period, assuming maximum profit, is calculated to be 2.72 years. This means the initial
investment can be recouped within this timeframe. This highlights the feasibility and financial viability of the
venture.
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Conclusion
Since the payback period is less than the required 3 years, it is recommended that the company should invest in
the boat tour business.
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Bibliography
Bostock, L. and Chandler, S. Mathematics. The Core Course for A-Level, United Kingdom: Stanley Thornes
(Publishers) Limited, 1997.
Campbell, E. Pure Mathematics for CAPE, Vol. 1, Jamaica: LMH Publishing Limited, 2007.
Hosein, R., and Gookool, R. Cape Economics Study Guide Unit 1, Caribbean Examinations Council, 2007
Martin, A., Brown, K., Rigby, P. and Ridley, S. Pure Mathematics, Cheltenham, United Kingdom: Stanley
Thornes (Publishers) Limited, 2000.
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