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?

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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KurtDavies KurtDavies · Answer 2 · 2024-10-01T14:44:25-04:00

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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