Two tanks of brine are connected via two tubes. Tank 1 contains x(t) pounds of salt in 500 gallons of brine, and tank 2 contains y(t) pounds of salt in 500 gallons of brine. Brine is pumped from tank 1 to tank 2 at a rate of 50 gallons/min and brine is pumped from tank 2 to tank 1 at the same rate of 50 gallons/min, ensuring that the total volume of brine in the tanks remains constant over time.

a) Show that x(t) and y(t) satisfy the differential equations x'= − (1/10)x + (1/10)y ; y' = (1/10)x − (1/10)y.

b) If initially tank 1 contains no salt, and tank 2 contains 10 pounds of salt, use the method of elimination to find x(t) and y(t).

a. Salt flows into tank A at a rate of

(y(t)/500 lb/gal) * (50 gal/min) = y(t)/10 lb/gal

and out at a rate of

(x(t)/500 lb/gal) * (50 gal/min) = x(t)/10 lb/gal

so the net rate of change of the amount of salt in tank A is

Similarly, you would find

b. We have

Notice that x' = -y', so

Solve for x: the characteristic equation

has roots and , so

Again using the fact that y' = -x', we then find

Given that x(0) = 0 and y(0) = 10, we find

find the magnitude of the trig term  

csc

(

x

)

−

1

by taking the absolute value of the coefficient.

1

find the lower bound of the range.

tap for more

y

=

−

2

find the upper bound of the range.

tap for more

y

=

0

the range is  

y

≤

−

2

or  

y

≥

0

.

(

−

∞

,

−

2

]

∪

[

0

,

∞

)

{

y

|

y

≤

−

2

,

y

≥

0

}

step-by-step explanation: