High School

A marine biologist measures the presence of a pollutant in an ocean and concludes that the concentration, C, in parts per million (ppm) as a function of the population, P, of the neighbouring town is given by C(P)=1.38 P+97.4. The population of the town, in thousands, can be modelled by P(t)=12(1.078)^(t) where t is the time in years since the first measurement.

a. Determine an equation, in simplified form, for the concentration of pollutant as a function of the number of years since the first measurement. [3 marks]

b. What reasonable restrictions should be placed on the function's domain and range? [2 marks]

c. The first measurement was taken in January 2018. Adapt the formula in part (a) to create an equation for the concentration as a function of the number of months since January 2020. [3 marks]

d. In which year will the concentration reach 180 ppm? [3 marks]

Answer :

Let's go through the problem step by step:

a. Determine an equation for the concentration of the pollutant as a function of the number of years since the first measurement.

  1. We know the concentration of the pollutant [tex]C[/tex] in ppm as a function of the population [tex]P[/tex] is given by:
    [tex]C(P) = 1.38P + 97.4[/tex]

  2. The population [tex]P[/tex] as a function of time [tex]t[/tex] in years is given by:
    [tex]P(t) = 12(1.078)^t[/tex]

  3. Substitute [tex]P(t)[/tex] into the equation for [tex]C(P)[/tex]:
    [tex]C(t) = 1.38(12(1.078)^t) + 97.4[/tex]
    [tex]= 16.56(1.078)^t + 97.4[/tex]

b. What reasonable restrictions should be placed on the function's domain and range?

  1. Domain:

    • [tex]t \geq 0[/tex]: Since [tex]t[/tex] represents time in years since the first measurement, it should be non-negative.

  2. Range:

    • [tex]C(t) \geq 97.4[/tex]: The concentration starts at 97.4 ppm and increases over time, so it will be at least 97.4 ppm.

c. Adapt the formula for the concentration as a function of the number of months since January 2020.

  1. Since the first measurement was in January 2018, by January 2020, 2 years have passed. Therefore, [tex]t = \frac{m}{12} + 2[/tex], where [tex]m[/tex] is the number of months since January 2020.

  2. Substitute this expression for [tex]t[/tex] back into the equation for [tex]C(t)[/tex]:
    [tex]C(m) = 16.56(1.078)^{(\frac{m}{12} + 2)} + 97.4[/tex]

d. In which year will the concentration reach 180 ppm?

  1. Set [tex]C(t) = 180[/tex]:
    [tex]16.56(1.078)^t + 97.4 = 180[/tex]
    [tex]16.56(1.078)^t = 82.6[/tex]
    [tex](1.078)^t = \frac{82.6}{16.56}[/tex]
    [tex](1.078)^t \approx 4.988[/tex]

  2. Taking the logarithm of both sides, we can solve for [tex]t[/tex]:
    [tex]t \approx \frac{\log(4.988)}{\log(1.078)}[/tex]

    • Solving this gives an approximate value for [tex]t[/tex].

  3. Calculate and round [tex]t[/tex] to the nearest year, which will give you the year in which the concentration reaches 180 ppm. Note that since [tex]t[/tex] is time since the first measurement in 2018, the final year will be 2018 plus the estimated number of years.

Other Questions