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A marine biologist measures the presence of a pollutant in an ocean and concludes that the concentration, [tex]C[/tex], in parts per million (ppm), as a function of the population, [tex]P[/tex], of the people who visit the beach is given by:

\[ C(P) = 1.38P + 97.4 \]

The population of people visiting the beach, in thousands, can be modeled by [tex]P(t)[/tex], where [tex]t[/tex] is the time in years since the first measurement:

\[ P(t) = 12 (1.078)^t \]

a. Determine an equation, in simplified form, for the concentration of pollutants as a function of the number of years since the first measurement.

b. How long (to the nearest year) will it take for the concentration to reach 180 ppm?

Answer :

Final answer:

The concentration of pollutants as a function of time since the first measurement is given by the equation C(t) = 1.38 * 12(1.078)^t + 97.4. It will take approximately 6 years for the concentration to reach 180 ppm.

Explanation:

The student's question involves finding the concentration of pollutants over time, as given by a mathematical function, and estimating the time it will take to reach a specific concentration level. By substituting the equation P(t) = 12(1.078)t into the equation C(P) = 1.38P + 97.4, we obtain the relationship C(t) = 1.38 * 12(1.078)t + 97.4.

To find the year in which the concentration reaches 180 ppm, we solve the equation 180 = 1.38 * 12(1.078)t + 97.4 for t. With some calculus, we find it will take approximately 6 years for the concentration to reach 180 ppm.

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Rewritten by : Barada

Final answer:

The equation for pollutant concentration as a function of time is C(t) = 16.56 * (1.078)^t + 97.4. It will take approximately 8 years for the concentration to reach 180 ppm.

Explanation:

We need to substitute the expression for P(t) into the equation for C(P). This results in C(t) = 1.38P(t) + 97.4 = 1.38 * 12 * (1.078)^t + 97.4. After simplifying, we get the equation C(t) = 16.56 * (1.078)^t + 97.4.

Next, for the concentration to reach 180 ppm, we solve the equation 180 = 16.56 * (1.078)^t + 97.4. Upon solving, we find t ≈ 8 years.

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