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Step-by-step explanation:

dA/dt = 6 − 0.02A

dA/dt = -0.02 (A − 300)

Separate the variables.

dA / (A − 300) = -0.02 dt

Integrate.

ln(A − 300) = -0.02t + C

Solve for A.

A − 300 = Ce^(-0.02t)

A = 300 + Ce^(-0.02t)

Use initial condition to find C.

50 = 300 + Ce^(-0.02 × 10)

50 = 300 + Ce^(-0.2)

-250 = Ce^(-0.2)

C = -250e^(0.2)

A = 300 − 250e^(0.2) e^(-0.02t)

A = 300 − 250e^(0.2 − 0.02t)

abidemiokin

Given the expression

  • [tex]\frac{dA}{dt} = 6 - 0.02A[/tex]

Factor out 0.02 from the xpression to have:

  • [tex]\frac{dA}{dt} = 0.02(300-A)[/tex]

Separate the variables;

  • [tex]\frac{dA}{300-A} = 0.02dt[/tex]

Integrate both sides of the eequation to have:

[tex]-ln(300-A) = 0.02t + C\\300-A = Ce^{-0.02t}\\A= 300 + Ce^{0.02t}\\[/tex]

Apply the initial condition to get the value of C

[tex]50=300+Ce^{-0.02(10)}\\50=300+Ce^{-0.2}\\-250=Ce^{-0.2}\\C = -250e^{0.2}[/tex]

Substitute the constant into the function A(t) to get the required proof

[tex]A(t)=300-250e^{0.2-0.02t[/tex]

Learn more on differential equations here; https://brainly.com/question/18760518