A tank originally holds 100 gal of water with 3 lb of salt dissolved. Then water containing LaTeX: \frac{1}{3}1 3lb of salt per gallon is poured into the tank at a rate of 5 gal/min. Well-mixed solution is leaving the tank at a rate of 1 gal/min. A. Find the initial-value problem that describes the amount of salt in the tank?B. Find the amount of salt in the tank after time T. C. If you have access to a computer with graphing capabilities, sketch the graph of solution foind in (b).

Let S(t ) denote the amount of salt in the tank at time t. Then S(0) = 3 lb.

Salt flows into the tank at a rate of

(1/3 lb/gal) * (5 gal/min) = 5/3 lb/min

and flows out at a rate of

(S(t)/V(t) lb/gal) * (1 gal/min)

where V(t) is the volume of liquid in the tank at time t. The tank starts off with 100 gal of liquid, and each min it gains a net volume of 5 - 1/3 = 14/3 gal of liquid, so that

V(t) = 100 + 14/3 t

Then the net rate of change of the amount of salt in the tank is governed by the linear differential equation,

dS/dt = 5/3 - S/(100 + 14/3 t)

or

dS/dt + 3S/(300 + 14t)= 5/3

To solve this ODE, multiply both sides by the integrating factor (300 + 14t)^(3/14), so that the left side can be condensed to the derivative of a product:

(300 + 14t)^(3/14) dS/dt + 3(300 + 14t)^(-11/14) S = 5/3 (300 + 14t)^(3/14)

d/dt [(300+14t)^(13/14) S] = 5/3 (300 + 14t)^(3/14)

Integrate both sides with respect to t, then solve for S:

(300+14t)^(13/14) S = 5/51 (300 + 14t)^(17/14) + C

S = 5/51 (300 + 14t)^(4/14) + C (300+14t)^(-13/14)

Given that S(0) = 3, we find

3 = 5/51 * 300^(4/14) + C * 300^(-13/14)

==>  C ≈ 498.99

Then the amount of salt in the tank at time t is given by

S(t) ≈ 5/51 (300 + 14t)^(4/14) + 498.99 (300+14t)^(-13/14)

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step-by-step explanation:

step-by-step explanation: the declaration of the rights of man and of the citizen (french: déclaration des droits de l'homme et du citoyen de 1789), set by france's national constituent assembly in 1789, is a human civil rights document from the french revolution.[1]

the declaration was drafted by the abbé sieyès and the marquis de lafayette, in consultation with thomas jefferson.[2] influenced by the doctrine of "natural right", the rights of man are held to be universal: valid at all times and in every place, pertaining to human nature itself. it became the basis for a nation of free individuals protected equally by the law. it is included in the beginning of the constitutions of both the fourth french republic (1946) and fifth republic (1958) and is still current. inspired by the enlightenment philosophers, the declaration was a core statement of the values of the french revolution and had a major impact on the development of freedom and democracy in europe and worldwide.[3]

the 1789 declaration, together with the 1215 magna carta, the 1689 english bill of rights (1689), the 1776 united states declaration of independence, and the 1789 united states bill of rights, inspired in large part the 1948 united nations universal declaration of human rights