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The amount of ozone, Q, in the atmosphere is decreasing at a rate proportional to the amount of ozone present. If time t is measured in years, the constant of proportionality is -0.0025. Write a differential equation for Q as a function of t.

Give the general solution for the differential equation. Let C represent the arbitrary constant from integration.
Q=C

If this rate continues, approximately what percent of the ozone in the atmosphere now will decay in the next 27 years? Round your answer to one decimal.

Respuesta :

Answer:

Differential equation: [tex]\frac{dQ}{dt} = rQ = -0.0025Q[/tex]

Solution of diff equation: [tex]Q(t) = Ke^{-0.0025t}[/tex]

6.3% of the ozone in the atmosphere now will decay in the next 27 years.

Step-by-step explanation:

The amount of ozone in the atmosphere may be found by the following differential equation:

[tex]\frac{dQ}{dt} = rQ[/tex]

In which r is the constant of proportionality and Q is the amount of ozone. A positive value of r means that the amount of ozone in the atmosphere is going to increase, while a negative value means it is going to decrease.

Solving the differential equation:

We integrate both sides of the differential equation and apply the exponential function. So:

[tex]\frac{dQ}{dt} = rQ[/tex]

[tex]\frac{dQ}{Q} = r dt[/tex]

Integrating both sides

[tex]\ln{Q} = rt + K[/tex]

Applying the exponential:

[tex]Q(t) = Ke^{rt}[/tex]

In which K is the initial amount of ozone.

So

[tex]Q(t) = Ke^{-0.0025t}[/tex]

If this rate continues, approximately what percent of the ozone in the atmosphere now will decay in the next 27 years?

This K-Q(27).

[tex]Q(27) = Ke^{-0.0025*27} = 0.9347K[/tex]

K - 0.9347K = 0.0653.

6.3% of the ozone in the atmosphere now will decay in the next 27 years.

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