Expert: Steve Holleran - 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
Hello Shames,

Okay, let's see what we can accomplish here.

(Q#2)  Since the equation is in the form y = a * e^bt,

and our original amount was 15,000, this is the "a" value.

So y = 15,000 * e^bt. When t = 2, y = 30,000, so

30,000 = 15,000 * e ^(2b), then

2   = e^(2b)   and 2b = ln 2 so b = ln 2 / 2.  Now the function becomes:

                  y = 15,000 * e^(ln 2/2 *t)

For t = 48, y = 15,000 * e^(ln 2/ 2 * 48)

             = 15,000 * e^(24 * ln 2)

             = 2.5165 * 10^11

For the average rate of change, I'm not sure what you want here.  Do you just want a general formula, or do you need the average rate from t = 2 to t = 48?  The average rate
would in any case be found in general, from time t1 to time t2 by:

delta y/delta t =
[15000e^(ln 2 / 2 * t1)-15000e^(ln 2/2 * t2)]/ (t1 - t2).


(Q#4)

If I2 = 2 * I1, then log(I2) = log(2*I1)

                            = log 2 + log(I1)

So the intensity is larger by a factor of log 2.

Don't know the physics of what's going on here, but since its logarithmic, it should increase greatly at first, then more slowly as time goes on.
It would be my guess that we tend to see objects not as bright as they are.

If AB = k * e^(I2/I1), and I2 = 2I1, then you have:

  AB = k * e^(2I1/I1) = k * e^(2).  

Originally, AB = k * e^(I1/I1) = k * e^1, so it has increased by a power of e.

When its tripled, AB = k * e^(3I1/I1) = k * e^(3), so compared to the original, it has increased by 2 powers of e.
Since this is exponential, I would say he perceives objects as being brighter than they are.

I'm somewhat shaky on the physics of the situation, but I believe what I've offered is the case.  I certainly hope I have helped, even if only a little.

Steve Holleran