Issue 001 - Public spending - Cost and budget
How big is U.S. pool maintenance compared with the Reflecting Pool project?
Recent reporting put the Lincoln Memorial Reflecting Pool repainting/waterproofing and algae-control work above $16 million. First estimate annual U.S. residential swimming pool maintenance spending. Then compare the public project with the annual upkeep for a number of backyard pools.
The problem
Estimate annual U.S. spending on cleaning and maintaining residential swimming pools. Then compare your estimate with the reported $16 million-plus Reflecting Pool project.
Part 1: national annual maintenance spending
Part 2: Reflecting Pool project equivalent
Because Fermi problems target an order of magnitude, I normally use no more than two significant digits and write most calculations in scientific notation; the Fermi reference explains both conventions.
Before checking sources
Matt's first pass
I started with approximately 1.3 x 10^8 U.S. households and assumed that 1 in 20 has a pool. I then assumed half of those pools are self-managed rather than serviced by a third party.
residential pools ~= (1.3 x 10^8 households) x (1 / 20)
~= 6.5 x 10^6 pools
professionally serviced pools ~= (6.5 x 10^6 pools) x 0.5
~= 3.3 x 10^6 poolsI assumed most serviced pools are open for only four months each year, while perhaps 20% are serviced year-round. I simplified that seasonal mix to about five $300 service visits per pool per year, or $1,500 annually.
annual service per pool ~= 5 visits/year x $300/visit
~= $1.5 x 10^3 per year
national service spending ~= (3.3 x 10^6 pools) x ($1.5 x 10^3/pool-year)
~= $5 x 10^9 per year
~= $5 billion per yearUsing that $1,500 annual cost, I initially calculated that the $16 million Reflecting Pool project corresponded to about 1,000 backyard pools maintained for a year.
my initial comparison:
$1.6 x 10^7 / $1.5 x 10^3 ~= 1 x 10^3 pool-yearsI also tried a per-gallon comparison from geometry. I imagined a round backyard pool about 6 feet in radius and 4 feet deep, which I rounded to a 2-meter radius and 1.3-meter depth.
pool volume ~= pi x (2 m)^2 x 1.3 m
~= 1.7 x 10^1 m^3
~= 1.7 x 10^4 liters
~= 4 x 10^3 gallonsAt about 4,000 gallons per pool, the 6.75-million-gallon Reflecting Pool would hold the same volume as roughly 1,700 backyard pools. Maintaining that many pools at $1,500 each would cost about $2.6 million annually, making the $16 million project about six times the annual service cost for an equal volume of residential pools.
volume-equivalent pools ~= (6.75 x 10^6 gal) / (4 x 10^3 gal/pool)
~= 1.7 x 10^3 pools
annual service ~= (1.7 x 10^3 pools) x ($1.5 x 10^3/pool)
~= $2.6 x 10^6
project ratio ~= $1.6 x 10^7 / $2.6 x 10^6
~= 6Calibration Score
Matt's Calibration Score: 60 / 100
Higher is better: earn points for accurate pegs, sound models, correct math, and a result close to the sourced answer. The image shows percent full of it: 100 minus the Calibration Score.
Pegs: 10/30. Pool ownership and average-pool-volume pegs were off, while annual maintenance cost was strong.
Model: 30/30. National pool count times annual maintenance was the right model.
Math: 0/10. The pool-years comparison included an order-of-magnitude arithmetic slip.
Result: 20/30. After correcting that slip, the estimate was good enough for Fermi work.
Grounding facts
The Reflecting Pool has historically held about 6.75 million gallons. A typical backyard pool is roughly 20,000 gallons, so the Reflecting Pool holds water comparable to about 300 to 350 typical pools.
volume comparison ~= 6.75 x 10^6 gallons / 2 x 10^4 gallons
~= 3.4 x 10^2
~= 340 typical poolsMatt's geometric estimate of about 4,000 gallons is internally sound for the small 12-foot-diameter, 4-foot-deep pool he pictured. The source-based 20,000-gallon peg represents a substantially larger backyard pool, which reduces the volume-equivalent count from about 1,700 pools to about 340.
That means the project cost is not simply scaling like a normal pool by water volume: 340 typical pools at $1,000 to $2,000 per year would suggest only about $340,000 to $680,000 of ordinary annual maintenance. On that basis, the $16 million project is roughly 24 to 47 times ordinary maintenance for the same water volume, rather than six times. The public project includes specialized landmark work, waterproofing, repainting, contracting, and algae-control systems.
After checking sources
Check and recalibrate
Start with a memory number: the United States has about 130 million households, or 1.3 x 10^8 households. If about 1 in 10 to 1 in 15 households has a pool, a reasonable starting point is about 10 million residential pools.
households ~= 1.3 x 10^8
pool ownership ~= 1 / 10 to 1 / 15
pools ~= (1.3 x 10^8) x (0.07 to 0.10)
~= 9 x 10^6 to 1.3 x 10^7
~= 1 x 10^7 poolsIf annual maintenance is roughly $1,000 to $2,000 per pool, national maintenance spending is:
national maintenance ~= (1 x 10^7 pools) x ($1 x 10^3 to $2 x 10^3 / pool-year)
~= $1 x 10^10 to $2 x 10^10
~= $10 billion to $20 billion per yearA broader pool-ownership definition that includes repairs, electricity, water, opening/closing, and other upkeep can push the range higher. For this problem, the cleaner target is recurring maintenance rather than all pool ownership costs.
Matt's $5 billion result is below this broader $10 billion to $20 billion range, largely because he counted only the half of pools assumed to receive professional service. It is still a successful order-of-magnitude estimate, and it exposes an important definition choice: professional service revenue is not the same quantity as all residential maintenance spending.
Reflecting Pool project ~= $1.6 x 10^7
backyard pool maintenance ~= $1 x 10^3 to $2 x 10^3 / pool-year
equivalent pool-years ~= ($1.6 x 10^7) / ($1 x 10^3 to $2 x 10^3)
~= 8 x 10^3 to 1.6 x 10^4 pool-yearsUsing Matt's own $1,500 assumption also gives about 11,000 pool-years. His written answer of 1,000 was a factor-of-ten arithmetic slip, not a problem with the estimation model:
$1.6 x 10^7 / $1.5 x 10^3
~= 1.1 x 10^4
~= 11,000 pool-yearsSo the reported Reflecting Pool project is closest to 10,000 pool-years of ordinary backyard-pool maintenance.
Post-check reflection
Matt's reflection
My assumption about pool ownership was off by about half, so that started things low. My annual maintenance cost was pretty spot-on, but then I made an order-of-magnitude error when figuring out how many pools the Reflecting Pool project corresponds to. Oops. Once that error is corrected, my guess was pretty good.
Stated as 10,000 pool-years of ordinary maintenance, the project sounds like a large amount to spend. But even before turning in my answer, I tried to correct for size by using a price-per-gallon estimate. That revealed my second weak intuition: how many gallons are in an average pool. I was way off. Correcting for that, the project does not feel ridiculous anymore, though maybe still a bit high.
Really, I should compare this with maintenance and repair costs for other similarly sized monuments to get a better sense of how much we should care about this number. Right now, it does not seem like too big of an issue.
Recommended memory peg
Remember approximately 1.3 x 10^8 U.S. households. It is a versatile starting point for estimating home ownership, utility use, consumer purchases, and residential infrastructure.
Reader results
Bars show how submitted estimates sort into the answer choices from the gut-check prompt.