Mathematical Applications #2

1) Many microorganisms experience brief periods of exponential growth. The formula for exponential growth is; N_t = N₀e^µt Where N₀ and N_t are the abundance of the organism(s) at time = 0 and later at time = t respectively, µ = the growth constant of the population per unit time, and t = the time that has elapsed since time = 0. If a bacterial population consisting of 10 cells experiences exponential growth for 48 hours, and has a growth constant (µ) value of 0.2 hr^-1, then what will be the value for N_t?

To answer this question you simply have to plug the values provided into the formula above to solve for N_t. The answer then becomes;

N_t = N₀e^µt --------> N_t = 10 e ^{(0.2h-1 * 48h)} --------> 10 e ^(9.6) --------> 1.476 x 10⁵ cells

New Question: If a initial bacterial population consisting of 10 cells experiences exponential growth for 48 hours, and has a growth constant (µ) value of 0.8 hr^-1, then what will be the value for N_t? What would be the value for N_t at 96 hours?

2) Light deceases rapidly with depth in the marine environment. The formula that describes this decrease is an exponential decline formula as follows;

I_z = I_o * e ^-kz

where I_z = light intensity at depth z, I_o = Light intensity at the surface, k = the light extinction coefficient, z = depth, and e is the exponential function. Using the empty plot below plot the light intensities at 25, 50, 75, 100, 250, and 500 meters if the light extinction coefficient is 0.014m^-1 and the surface light intensity is 1100 µE cm2 s.

Notice the similarity between this formula and the formula used in question 1. The formula above is a negative exponential (because of the negative value within the exponential function) meaning that with depth the light intensity will decrease rapidly. Again here you plug the values provided into the formula above to solve for I_z. Determining the values for I_z in the table below;

Plotting this data on the empty figure provided we get;

New Question:   Using the empty plot above plot the light intensities at 0, 50, 100, 200, 300, and 500 meters if the light extinction coefficient is 0.019m^-1 and the surface light intensity is 1200 µE cm2 s.  Provide the e3xact values for these depths also in a list.

3) You have just graduated from the Acme School of Oceanography and have landed your first job as a zooplankton ecologist. You are to study two important zooplankton species that vertically migrate on an upcoming oceanographic cruise. Species "A" migrates upwards from 300m at a rate of 0.8m / minute, while Species "B" migrates upwards from 420 m at a rate of 1.2m / min. Both Species begin their upward migration at dusk (2000hrs mt).

a) When will these species be at the same depth as they rise to the surface. Set up this algebraic problem in the form of two linear equations (i.e. y = mx ± b) that are equal to each other, and present your results.

b) If you want to obtain both species at the depth indicated from answer a), how much trawl line should you pay out to reach that depth, if the trawl line extends off the stern at a 60° angle relative to the surface water. (Hint: this is a problem involving the applications of the Pythagorean theorem of geometry)

To answer a above you need to express the movements of the two zooplankton species as linear equations. For each species this can be illustrated below where y = depth at time x, m = movement rate, x = time, and b = maximum depth;

Species A: y = (0.8m min-1*(time)) - 300 m

Species B: y = (1.2m min-1*(time)) - 420 m

Setting these equations equal to each other we have;

(0.8m min-1*(time)) - 300 m = (1.2m min-1*(time)) - 420 m, now solving for time we get;

(0.8m min-1*(time)) = (1.2m min-1*(time)) - 120 m,

-0.4m min-1*(time) = -120 m,

time = -120 m / -0.4m min-1,

time = 300 minutes

For part b we use the information obtained in part a, namely the depth after 300 minutes of migration, which is 60 meters. An illustration helps to understand this problem.

So to determine the ? side dimension knowing the angles indicated we can use simple geometry. To review the relationships between sides and angles in a right triangle you can visit the following web page (click here), but the answer to are question is below;

sin 60^o = 60 m / ? m

0.866 = 60 m / ? m

? = 69.28 m

New Question:  You have just graduated from the Acme School of Oceanography and have landed your first job as a zooplankton ecologist. You are to study two important zooplankton species that vertically migrate on an upcoming oceanographic cruise. Species "A" migrates upwards from 400m at a rate of 1.2m / minute, while Species "B" migrates upwards from 520 m at a rate of 1.6m / min. Both Species begin their upward migration at dusk (1800hrs military time).

a) When will these species be at the same depth as they rise to the surface. Set up this algebraic problem in the form of two linear equations (i.e. y = mx ± b) that are equal to each other, and present your results.

b) If you want to obtain both species at the depth indicated from answer a), how much trawl line should you pay out to reach that depth, if the trawl line extends off the stern at a 30° angle relative to the surface water. (Hint: this is a problem involving the applications of the Pythagorean theorem of geometry)