Jonathan Thomson's web journal

How Big is Lily Pad Lake? November 4, 2012

Filed under: Uncategorized — jethomson @ 8:53 pm

Here’s a common brain teaser:

“In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?”

 

Many people answer a bit too quickly and say 28 days. However, the correct answer is 47 days assuming the doubling on the 48th day doesn’t overshoot the area of the lake. On the 47th day the lily pads cover half of the lake and on the 48th day they double covering the entire lake. But what I’m really interested in is how big is Lily Pad Lake?

 

Let’s denote the area of lily pad Lake by A.

 

Then the area covered by the lily pads on the 48th day is A. Therefore the formula for determining the area covered by the lily pads on day n is:

 

On the 48th day Alily is equal to A, and on the 0th day Alily = A/(2^48). The problem doesn’t state how many lily pads are present on the 0th day, but to minimize the size of Lily Pad Lake I’ll assume just one lily pad is present. Thus on the 0th day Alily is equal to the area of 1 lily pad. Assuming the area covered by one lily pad is 5 square feet, then:

 

That’s 1.4 quadrillion square feet or 50,482,628 square miles or 130,749,410 square kilometers. For comparison purposes the Pacific Ocean is 155,557,000 square kilometers and the next biggest ocean is the Atlantic at 76,762,000 square kilometers. So perhaps a better name for Lily Pad Lake would be Lily Pad Ocean, which would have to be a freshwater ocean since lily pads don’t live in salt water.

 

 

Notes
Typical [lily pad] plant size 5-10′ square

The Pacific Ocean is 155,557,000 sq km

 

Evaluating an Off-the-shelf Actinic Bulb for use in Blue Light Therapy Acne Treatment

Filed under: Spectrometer — jethomson @ 7:48 pm

 

There’s been some research showing that light at around 415 nm is effective at killing the bacteria that cause acne. This type of acne treatment is know as blue light therapy although the light in the region around 415 nm is actually violet. The British paper “Phototherapy with blue (415 nm) and red (660 nm) light in the treatment of acne vulgaris” by P. Papageorgiou, A. Katsambas, and A. Chu states in its Materials and Methods section that the blue light source they used “had an asymmetric peak of 415 nm +20/-15 nm” and an irradiance of 4.23 mW/cm^2 at 25 cm (about 10 inches). Actinic lamps produce blue light with a peak irradiance around 415 nm. I used my homemade spectrometer to capture the actinic spectrum of a Coralife 50/50 20W compact fluorescent. In the range of wavelengths from 400 to 430 nm I measured a total irradiance of approximately 0.7 mW/cm^2 at 3 inches (7.65 cm). Therefore the Coralife actinic lamp has only a sixth of the strength of the lamp used in the British paper. However, the experiment used an irradiance time of 15 minutes such that the patient was exposed to a “cumulative dose of 320 J/cm^2″. Since the Coralife actinic lamp is a sixth power of the experimenter’s lamp, then multiplying the exposure time by six should result in the same dosage. Therefore one could hypothesize that an exposure time of 90 minutes with a Coralife 20W lamp will produce similar results to those seen in the paper. I haven’t tested this hypothesis because 3 inches is quite a small distance and having the lamp that close would prevent me from doing anything else during the therapy session. The lamp would obstruct my field of view and the blue light is an eye hazard. Given that it would take 90 minutes to receive a full dose and that a portion of the light output is in the UV wavelengths makes blue light therapy by means of a Coralife bulb even less attractive. However, it might be possible to spread the exposure time over multiple sessions and use those sessions to rest or listen to music. If anyone tries this out please leave a comment.

 

 
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