Find the missing length the triangles are similar

Length of the missing side is 13

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(A) Let A(t) denote the amount (in grams, g) of the chemical present in the pond at time t (in hours, h). The starting amount is unknown; call it a, measured in g, so that A(0) = a.

Water containing 0.01 g/gal of the chemical flows in at a rate of 300 gal/h, which increases the amount of the chemical in the pond at a rate of

(0.01 g/gal) • (300 gal/h) = 3 g/h

and flows out at the same rate, so the amount decreases by

(A(t)/1,000,000 g/gal) • (300 gal/h) = 3A(t)/10,000 g/h

Then the net rate of change of the amount of chemical in the pond is given by the ODE,

dA(t)/dt = 3 - 3/10,000 A(t)

(B) Solve the ODE for A(t). There are several ways to do that. For instance, itt's separable, so we have

dA(t) / (3 - 3/10,000 A(t)) = dt

Integrate both sides to get

-10,000/3 ln|3 - 3/10,000 A(t)| = t + C

Solve for A(t) :

ln|3 - 3/10,000 A(t)| = -3/10,000 t + C

3 - 3/10,000 A(t) = exp(-3/10,000 t + C )

3 - 3/10,000 A(t) = exp(-3/10,000 t ) • exp(C )

3 - 3/10,000 A(t) = C exp(-3/10,000 t )

3/10,000 A(t) = 3 - C exp(-3/10,000 t )

A(t) = 10,000 - C exp(-3/10,000 t )

Since A(0) = a, we have

a = 10,000 - C exp(-3/10,000•0)   →   C = 10,000 - a

→   A(t) = 10,000 - (10,000 - a) exp(-3/10,000 t )

As t grows to infinity, the exponential term will vanish, leaving a limiting amount of 10,000 g of the undesirable chemical in the pond, which does not depend on the original amount.