Sunday, May 12, 2013

Applications of Differential Equations (4): More Word Problems

First let me give a jump start assist for Problem No. 1 in Application of DEs (3). The Ship Problem. With this jump start most math mavens ought to be able to finish the solution!

We show:  Thrust = ma = 200,000 lbs. = kv at outset

Afterward with velocity v:    200,000 lb. = 10,000 v

So: v = 200,000 lbs./ 10,000 ft-lbs./sec = 20 ft/sec

We now need a function of time, so first write:

mg – ma = kvn

Or: m(g – a) =  kvn

Then:  (g – a) = (k/ m) vn

Let a = dv/dt and let (k/m) = a

Then write the differential eqn.: (g – dv/dt) =  a vn

Separate variables to get:

dv/ (g -  a vn)  = dt  

The relation is linear (constant propeller thrust) so take exponent n = 1:

Þ     dv/ (g -  a v)  = dt  

Integrate the above to obtain:

 ln (g -  a v)    = -at + C   

(Further Hint: You will need to obtain the value of  a  to get the integrating factor in terms of exp (-at))

Answers to problems (2) and (3):

(2) T =   p/4 s  and f =  4/ p s-1

(3) 177.7 lbs.

Additional Differential equation word problems:

1. A ball weighing ¾ lb. is thrown vertically upward from a point 6 feet above the surface of the earth with an initial velocity of 20 feet per second. As it rises it is acted upon by air resistance which is numerically equal to (1/64) v in pounds, where v is the velocity in ft /sec. How high will the ball rise?

2. The rate at which radioactive nuclei decay is proportional to the number of such nuclei that are present in a given sample. Half of the original number of radioactive nuclei have undergone disintegration after 1500 years.

a) What percentage of the original sample will remain after 4500 years?

b) In how many years will only one tenth of the original number remain?

(Hint: see )

3. The population of a city increases at a rate proportional to the number of its inhabitants present at any time t. If the city’s population was 30,000 in 1950  and 35,000 in 1960, what would its population be in 1970? What would it be in 2020?

4. In a certain bacteria culture the rate of increase in the number of bacteria is proportional to the number present.

a) If the number triples in three hours, how many will be present in 10 hours?

b) When will the number be ten time the number initially present?

5. Newton’s law of cooling states that the rate at which a body cools is proportional to the difference between the temperature of the body and that of the medium in which it is situated. Assume a body at temperature T = 80F is placed at time t = 0 in a medium which has a temperature maintained at 50F. At the end of five minutes the temperature of the body has cooled to 70F.

a) Find the temperature of the body at the end of 10 minutes?

b) When will the temperature of the body be 60F?

-----Answer Key to Selected Problems :

1.Rises 55.08 ft, or 55.08 ft + 6 ft = 61.08 ft. relative to ground

2. (a) One-eighth or 12.5% of the original number remains after 4500 yrs.

(b) t = 4985 yrs.

3. 40, 833 by 1970

5. (a) 63.33 F  

(b)  13.55 mins.

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