Solve using dirac deltla/discontinuous forcing
consider the following scenario. a salt tank that initially contains 50 gallons of pure water. a brine solution containing 1/5 lb/gal of salt flows in to the tank at a rate of 5 gal/min. brine flows out of the tank at the same rate. at time t = 15 minutes the mechanism regulating salt flow in to the tank breaks and 20 pounds of salt is dumped instantaneously in to the tank. although water continues to flow in to the tank at the original 5 gal/min, there is no salt in the water.
1. write an ivp describing the amount of salt in the tank at time t.
2. solve the ivp from problem 1.
3. plot the solution found in problem 2 and explain the results.

Let denote the amount of salt in the tank at time . We're told that .

For , the salt flows in at a rate of (1/5 lb/gal)*(5 gal/min) = 1 lb/min. When the regulating mechanism fails, 20 lbs of salt is dumped and no more salt flows for . We can capture this in terms of the unit step function and Dirac delta function as

(in lb/min)

The salt from the mixed solution flows out at a rate of

Then the amount of salt in the tank at time changes according to

Let denote the Laplace transform of , . Take the transform of both sides to get

Solve for , then take the inverse of both sides.

unless if u could do my questions, if not i wouldn't want to

the largest number of the four options is 1.8 x 10-2

you have it in scientific notation.

1.80 x 10-6   = 0.00000180

1.80 x 10-4   = 0.000180 (bigger, or "less small", than the former one)

so

1.8 x 10-2       = 0.0180. as you see. the "less small", or the biggest one of them all.

as you have negative exponent, it means you are dividing 1.8 by 100 (i.e. moving the dot towards left, two times).

if you had positive exponent for the scientific notation, power of tenths, you would be multiplying times 100 (moving the dot towards right) as below:

1.8 x (10)2 = 180

so. if you have negative exponent in scientific notation, move the dot toward left. if it is positive, toward right as many times the exponent tells you.