Expert: Paul Klarreich - 5/4/2007

Question
please listen this is a project and i solved arond 10 pages and these are the questions i faced some problem solving so please help me i need your knowledge

(Q#2) A certain type of primitive bacteria was  recently discovered in a rock sample from a meteor shower. The original count was 15000 bacteria. Placed in a nutrient rich rich environment in a government research lab in Burbank, California, the bacteria began to grow rapidly. Two hours later, the number of bacteria had doubled, that is, there were 30000 bacteria in the culture. Effort to slow the growth of the bacteria failed and after two days the entire lab was tangled mass of weird, pulsating alien slime. Use the exponential growth model ( y=ae^(bt) ) B>0.

1)   find an equation that represents the number of bacteria ( t ) hours after the original count of 15000.
2)   Find the amount of bacteria present after 48 hours.
3)   What is the average growth rate?


(Q#3) Logarithmic scales: (sounds fishy to me!) if you study some of the results from experimental psychology, it turns out that human senses such as sight and hearing operate on a logarithmic scale. (true fact !) for example, suppose you are staring at a light bulb that gives off light measure at a certain intensity ( I ). Someone turns the switch and suddenly the light intensity emitted by the bulb is twice as much as the first intensity. What do you perceive has happened? That is, does the light look twice as bright? No! to the average person, it only looks on the order of (log2 times I)as bright! Our “visual information processing” system changes (transduces) the incoming light signals.  

1)   if you double the intensity of the light bulb, that is, increase it to 2 x I, what does the average person perceive (roughly speaking) has happened? (how much brighter does the light bulb look? )
2)   since perception brightness is logarithmic, would you say that we tend to see lighted objects brighter than they really are, or not as bright?


(Q#4) (Refer back to #3 above.) suppose that the person viewing the light is a vampire. Lets say that to a vampire, perceived light intensity looks different than it does to us mortals. Vampires allegedly having keen eyesight (and hearing as well), let us say that the vampires perception works exponentially rather than logarithmically. That is, when you turn up the lights from intensity (I#1) to intensity (I#2), the change is perceived brightness ÄB is found by the formula,

           
ÄB=ke^(I#2/I#1)

1)   what happens when the actual light intensity is doubled? By how much is perceived brightness magnified?
2)   What about when the actual light intensity is three times greater? By how much is the perceived brightness magnified?
3)   Since the vampire perception of light is (according to our light of fancy here) exponential, would you say that he tends to see lighted objects brighter than they really are, or not as bright?


Answer
Questioner:   shames
Category:  Advanced Math
 
Subject:  logarithims and exponents
Question:  please listen this is a project and i solved aroond 10 pages and these are the questions i faced some problem solving so please help me i need your knowledge

(Q#2) A certain type of primitive bacteria was  recently discovered in a rock sample from a meteor shower. The original count was 15000 bacteria. Placed in a nutrient rich rich environment in a government research lab in Burbank, California,

>> Obviously your teacher was a fan of Johnny Carson.

the bacteria began to grow rapidly. Two hours later, the number of bacteria had doubled, that is, there were 30000 bacteria in the culture.

Effort to slow the growth of the bacteria failed and after two days the entire lab was tangled mass of weird, pulsating alien slime.

>> You got this from a 1960's movie, right?

Use the exponential growth model ( y=ae^(bt) ) B>0.

1) find an equation that represents the number of bacteria ( t ) hours after the original count of 15000.
2) Find the amount of bacteria present after 48 hours.
3) What is the average growth rate?

>> -------------------
Using  N = N0 e^kt   (same as yours but slightly different notation.)

The bacteria doubled in 2 hours.  So:

2N0 = N0 e^k(2), where N0 was (is?) your 15000 original bacteria.

2 = e^2k
2k = ln 2
k = 0.5 ln 2

After 48 hours, since the doubling time is 2 hours, the bacteria have doubled 24 times, so the number of bacteria is:

N = N0 (2^24)
 = a lot.

I am not sure what you mean by 'average growth rate', but a reasonable interpretation is:
     N(48) - N(0)
AGR = ------------
       48 - 0

You can work out the arithmetic.

>> ------------------------------

(Q#3) Logarithmic scales: (sounds fishy to me!)

>> Just a minute, here!  As the expert, I do the funnies, please.

if you study some of the results from experimental psychology, it turns out that human senses such as sight and hearing operate on a logarithmic scale. (true fact !) for example, suppose you are staring at a light bulb that gives off light measure at a certain intensity ( I ). Someone turns the switch and suddenly the light intensity emitted by the bulb is twice as much as the first intensity. What do you perceive has happened? That is, does the light look twice as bright?

No! to the average person, it only looks on the order of (log2 times I)as bright! Our “visual information processing” system changes (transduces) the incoming light signals.  

>> I don't think this is really well-defined (nothing from psychologists ever is), but I'll see what I can do.

1) if you double the intensity of the light bulb, that is, increase it to 2 x I, what does the average person perceive (roughly speaking) has happened? (how much brighter does the light bulb look? )

>> Not so clear, but perhaps an increase by a factor of  1 + ln 2 = 1.69 might do it.

2) since perception brightness is logarithmic, would you say that we tend to see lighted objects brighter than they really are, or not as bright?

>> Yes.  (Well, you asked.)


(Q#4) (Refer back to #3 above.) suppose that the person viewing the light is a vampire. Lets say that to a vampire, perceived light intensity looks different than it does to us mortals. Vampires

allegedly having keen eyesight (and hearing as well), let us say that the vampires perception works exponentially rather than logarithmically. That is, when you turn up the lights from intensity (I#1) to intensity (I#2), the change is perceived brightness ÄB is found by the formula,

          
ÄB=ke^(I#2/I#1)

>> I think you probably mean:  AB = e^(I1/I2)

1) what happens when the actual light intensity is doubled? By how much is perceived brightness magnified?

>> That would mean  AB = k e^2

2) What about when the actual light intensity is three times greater? By how much is the perceived brightness magnified?

3) Since the vampire perception of light is (according to our light of fancy here) exponential, would you say that he tends to see lighted objects brighter than they really are, or not as bright?

>> See earlier answer.