### The Economics of the 6 Second Toothbrush

A little while ago Mashable did a story on Blizzident, a 3D-printed toothbrush that claims to effectively clean your teeth in just 6 seconds. I was thinking about how efficient this would be, but was a little put out by the price ($339 for a brush that lasts a year, not including the cost of getting your mouth scanned to make it).

So I decided to do a little math and see if I could figure out if it was worth it. This calculation makes a few assumptions:

- You brush your teeth for two minutes twice a day (which I totally do…)
- Time not spent brushing teeth would be spent doing something somewhat productive

## Cost of Conventional Brushing:

Cost of Toothbrush: $10

Cost of Time:

if H = hourly wage ($/hr) Time Spent Brushing per day = 4 min or 1/15 hours Time Spent Brushing per year = (1/15 hours/day) x (365 days/year) = 73/3 = 24.33 hrs/year Cost of Time = H x (73/3)

Total Yearly Cost:

Cost = 10 + 73H/3

## Cost of Blizzident Brushing:

The cost of the scan needed to make the Blizzident varies. According to their website, the cost to get your teeth 3D scanned or to have an impression made and then send it to a place where they will 3D scan it costs between $75 and $250, which is quite a large range. For simplicity, let’s just use the average: $162.5.

Cost of Toothbrush:

Total Cost ( $) = 339 + 162.5 = 501.5

Cost of Time:

if H = hourly wage ($/hr) Time Spent Brushing per day = 12 sec or .2 min or .2/60 hrs Time Spent Brushing per year = 365(.2/60) = 73/60 hrs Cost of Time = H x (73/60)

Total Yearly Cost:

Cost = 501.5 + (73/60)(H)

## Comparison:

I’ve ignored one thing; the Blizzident has to be replaced yearly (as does, I assume, your regular toothbrush) and the replacement Blizzident is cheaper than the original. A Blizzident can either be refurbished (for $89) or replaced (for $159).

#### Cost Comparison for Year 1:

10 + (H)(73/3) = 501.5 + (H)(73/60) H(73/3) - H(73/60) = 491.5 H (73/3 - 73/60) = 491.50 H = 21.26 $/hr

So it makes sense to buy a Blizzident if you make more than $21.26/hour.

#### Cost Comparison for Multiple Years :

let t = the number of years one uses the Blizzident or the conventional toothbrush

Cost of Conventional Brush for t years:

t (10 + H(73/3)) = 10t + (H)(t)(73/3)

Cost of Blizzident Brush for t years if replacing with refurbished brush:

501.5 + (t-1)(89) + 73tH/60 = 412.5 + 89(t) + (t)(H)(73/60)

Cost of Blizzident Brush for t years if replacing with new brush:

501.5 + (t-1)(159) + 73tH/60 = 342.5 + 159(t) + (t)(H)(73/60)

`Thus the comparison of a refurbished-replaced Blizzident with a conventional brush is: `

412.5 + 89(t) + (t)(H)(73/60) = 10t + (H)(t)(73/3) 412.5 + 79(t) = (t)(H)(73/3 - 73/60) 412.5 + 79(t) = (t)(H)(1387/60)

`So for the 1st year: `

412.5 + 79 = (1387/60)(H) 21.26 = H

`For the 2nd year: `

412.5 + 79(2) = (2)(H)(1387/60) 12.34 = H

`For the 3rd year : `

412.5 + 79(3) = (3)(H)(1387/60) 9.37 = H

And the comparison of a new-replaced Blizzident with a conventional brush is:

342.5 + 159(t) + (t)(H)(73/60) = 10t + (H)(t)(73/3) 342.5 + 149(t) = (t)(H)(73/3 - 73/60) 342.5 + 149(t) = (t)(H)(1387/60)

`So for the 1st year: `

342.5 + 149(1) = (1)(H)(1387/60) 21.26 = H

`For the 2nd year: `

342.5 + 149(2) = (2)(H)(1387/60) 13.85 = H

`For the 3rd year: `

342.5 + 149(3) = (3)(H)(1387/60) 11.38 = H

So it makes sense to buy a Blizzident if you make more than $21.24/hour.

## Results:

Buying a Blizzident saves you money the first year if you make more than $21.24 per hour.

Buying a Blizzident and then refurbishing it yearly after the first year (for two years) makes sense if you make more than $12.34 per hour. Doing so for three years makes sense if you make more than $9.37 per hour.

Buying a Blizzident and then replacing it yearly after the first year (for two years) makes sense if you make more than $13.85 per hour. Doing so for three years makes sense if you make more than $11.38 per hour.