A couple weeks ago I posted this problem. I like the problem, but I wasn’t a huge fan of its solution and frankly, it sort of got away from me. Thankfully, my colleagues are more adept than I am at developing clear problems. So here’s the updated, improved version.
===============================================
Moving On Up
Artifact / Entry Event
Facilitator : “Great news everybody. Principal ________ gave me a huge raise. It turns out that, contrary to all the media reports, the school district is flush with cash. As such, I’m moving on up. To the East side. I bought a MTV-cribs-like house. Unfortunately, I don’t have my hands on the huge sums of money yet, so I’ll be using a U-Haul to transport my currently meager possessions. Here’s a map of my old house, new house, and U-Haul rental location.”
[Facilitator shows slide / handout of map]
Facilitator: “Now, the local U-Haul rental has different fees for different sized trucks. The larger the moving truck, the more it costs. Here is their pricing scheme.”
[Facilitator shows slide / handout of U-Haul pricing table. Note: alternatively, the facilitator could direct them to the U-Haul website to obtain the specs: 14’ truck and 20’ truck. Note: there is also a 17’ truck for an up front cost of $29.95.]
Truck length | U-Haul listed capacity | Up front cost (1 day rental) | Additional cost per mile |
14’ | 733 cu. Ft. | $19.95 | $0.79 |
20’ | 1015 cu. Ft. | $39.95 | $0.79 |
Facilitator: “Now, I’m not exactly sure how much stuff I have. It might take 1 trip with these trucks, it might take 10. So I’m going to go calculate my furniture while you guys help me come up with a mathematical model, suggesting which truck I should get depending on the number of trips I have to take.”
Potential solution:
Students should first come up with a basic functional representation of the cost vs. mileage before applying it to the number of trips.
Cost = (up front cost of truck)+$0.79*mileage
, where m=mileage driven. This can be graphed as well.
To make a choice on which truck to get, students will have to determine a distance to and from each location on the map.
From U-Haul rental to Old Home: ~5.5 mi
From Old Home to New Home: ~8.5 mi
From New Home to U-Haul Rental: ~9 mi
Students then will need to set up a model in the form of a linear function, depending on the number of trips between the Old Home and New Home. Each extra trip after the first one will tack on 8.5+8.5=17 miles to the total mileage.
Cost = (Up front cost of truck)+$0.79*(mileage from U-Haul to Old Home + mileage between Old Home and New Home + mileage from New Home to U-Haul) + $0.79*2*(mileage from Old Home to New Home)*(number of trips – 1)
Cost = (Up front cost of truck)+$0.79*(5.5+8.5+9) + $0.79*2*(8.5)*(number of trips -1)
Cost = (Up front cost of truck)+$0.79*(23) + $0.79*2*(8.5)*(number of trips – 1)
Cost = (Up front cost of truck)+$18.17+$13.43*(number of trips – 1)
We can break that down for the three given truck sizes. For the smallest truck (the 14’ truck):
where x = # of trips to and from the New/Old Home
For the other two trucks:
Using a table:
# of trips: |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
14’ truck |
38.12 |
51.55 |
64.98 |
78.41 |
91.84 |
105.27 |
118.7 |
132.13 |
145.56 |
158.99 |
20’ truck |
58.12 |
71.55 |
84.98 |
98.41 |
111.84 |
125.27 |
138.7 |
152.13 |
165.56 |
178.99 |
Entry Event #2
Facilitator: “I’m still working on figuring out the number of trips I’ll have to take. I had several moving companies come over – before I decided to rent a U-Haul – and they each gave me a wildly different estimate of the amount of cubic footage of stuff I have.
Estimate 1: 2225 cu. ft |
Estimate 2: 2891 cu. Ft. |
Estimate 3: 3262 cu ft. |
Which size truck should I rent, based on each estimate?”
Potential solution.
Each estimate would entail a different number of trips based on the size of the truck rented. Recall,
Truck length | U-Haul listed capacity |
14’ | 733 cu. Ft. |
20’ | 1015 cu. Ft. |
Truck capacity |
14’ truck (733 cu ft) |
20’ truck (1015 cu ft) |
Estimate 1: 2025 |
3 trips |
2 trips |
Estimate 2: 2891 |
4 trips |
3 trips |
Estimate 3: 3462 |
5 trips |
4 trips |
We can translate this into cost also using a table. (most cost effective)
Truck capacity |
14’ truck (733 cu ft) |
20’ truck (1015 cu ft) |
Estimate 1: 2025 |
64.98 |
51.55 |
Estimate 2: 2891 |
78.41 |
74.98 |
Estimate 3: 3462 |
91.84 |
98.41 |
So for the three estimates, the first two would be more cost efficient using the 20’ truck while the last estimate would be more cost efficient using the 14’ truck.
Extension questions:
Can we develop a rule of thumb for differentiating between getting a 14’ vs. a 20’ truck (i.e. “If the # of trips using the 14’ truck is two or greater than the 20’ truck then it’s more cost efficient to get the 20’ truck.)?
Can we calculate the actual square footage of the 14’ and 17’ trucks? This incorporates volume standards.
======================================
Here’s a downloadable pdf version of the above activity.
======================================
I still am not 100% sure this activity is ready to go. I feel like it could still be improved. By all means, if you have any suggestions, let me know in the comments.
4 thoughts on “U-haul Linear Systems problem (updated and improved)”