Description
questions 2,3,4,5,6. ً
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Explanation & Answer
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Problem #2
Given:
Rental company
I
II
III
Daily rate
$23.50
$19.99
$15.50
Mileage rate
0
$0.15
$0.22
Question:
How should one decide which rental car company to rent from for a 7 day rental?
Solution:
First, Polya’s problem solving method is based on the following:
1.
2.
3.
4.
Understanding the problem
Planning the steps in solving the problem
Implementing the steps and strategies to solve the problem
Verifying the solution
The problem statement
To understand the problem statement, we ask a series of questions:
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Are all the words clear and the terms defined and understood?
Can we restate the problem in our own words?
Does the problem make sense?
Is the problem like other problems that we might have worked on?
Do we understand the goal of the problem?
And others
For this problem, we see that:
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We are asked to choose the best company to rent from
We choose what the basis for “best” is. And it’s probably cost.
The rates are crystal clear (as seen in that table)
We see that there are 2 differences. Daily rates and mileage, the lower the daily rate,
the more per mile.
The steps or process for setting up the solution
To setup the solution, we think along the following lines in the Polya process:
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We decide what to base our decision on (our metric)
We look for patterns in the data
We think about what equations we might setup and use
We think about how to manipulate those equations
Along the way, we consider graphs, images, linear algebra
•
We think about the tricks and tools we’ve learned and practiced and how they might
apply.
For this problem, we’re going to follow these steps
1.
2.
3.
4.
We choose total cost as our metric we base our decision on
We derive equations to generate total cost for each company
We’re going to visualize the lowest rate by plotting all three total costs on one plot
From the, plot, we can see which intersections we will need to resolve to generate
ranges
5. Finally, we’ll report the lowest cost for each range
Implementing the steps
In the Polya process, we implement the solution by:
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We follow the path we outlined previously
If the path doesn’t lead us to a solution, we vary it and try again.
We lookup any additional data or tools we might need along the way
We restart as needed
For this problem, we see that
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We generate the total costs as shown in the table below
We observe the equations are linear with different y intercepts (based on different daily
costs) and different slopes (based on different mileage rates)
We plot all three equations on the same plot with mileage on the x-axis and cost on the
y-axis.
We identify transition points on the plot (point A for example)
We read from the plot, which companies are least expensive
We identify the x-values (mileage) for the transition point(s).
Rental company
I
II
III
Daily rate
$23.50
$19.99
$15.50
Mileage rate
0
$0.15
$0.22
Total cost
$23.50*7 + $0.00*miles = $164.50
$19.99*7 + $0.15*miles
$15.50*7 + $0.22*miles
•
•
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Now we can easily see company III is cheapest from x = 0 miles to x = the x-coordinate of
point A and we can easily see that company I is cheapest if miles exceeds that value.
We can now determine the x value that corresponds with point A
We know that the lines for company I and company III intersect at point A so we can
write
𝑡𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝐼 = 𝑡𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝐼𝐼𝐼
$164.50 = $15.50 ∗ 7 + $0.22 ∗ 𝑚𝑖𝑙𝑒𝑠 𝑑𝑟𝑖𝑣𝑒𝑛
𝑚𝑖𝑙𝑒𝑠 𝑑𝑟𝑖𝑣𝑒𝑛 =
•
$164.50 − $15.50 ∗ 7
= 254.54 𝑚𝑖𝑙𝑒𝑠
$0.22 𝑝𝑒𝑟 𝑚𝑖𝑙𝑒
Meaning, rent from company III if planning on driving less 254.54 miles and from
company I if driving more than 254.54 miles
Verification of the solution
In general, we verify the solution
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Answers the question or problem
Is accurate / correct / precise.
Makes sense and is the most direct solution
Can be generalized for other problems
For this problem, we see that
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We did answer the question.
The solution is direct and the logic makes sense
We can generalize and even suggest alternative methods if say days rented varies
Is accurate (see the table below yellow highlight is lowest cost)
Miles driven
0
50
100
150
200
250
300
350
400
450
500
550
Company I
164.50
$164.50
$164.50
$164.50
$164.50
$164.50
$164.50
$164.50
$164.50
$164.50
$164.50
$164.50
Company II
$139.93
$147.43
$154.93
$162.43
$169.93
$177.43
$184.93
$192.43
$199.93
$207.43
$214.93
$222.43
Company III
$108.50
$119.50
$130.50
$141.50
$152.50
$163.50
$174.50
$185.50
$196.50
$207.50
$218.50
$229.50
Problem #3
Given:
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On 9.21.2011, the US debt = $14,705,188,992
The interest on that debt = 3% / yr compounded monthly
Question:
How much should the government pay each month from May to December to bring that debt to
$7 trillion?
Solution:
Again, we’re going to use Polya’s method
1.
2.
3.
4.
Understanding the problem
Planning the steps in solving the problem
Implementing the steps and strategies to solve the problem
Verifying the solutio...
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