This exercise uses the population growth model.

Bacterium Rhodobacter sphaeroides

A certain culture of the bacterium Rhodobacter sphaeroides initially has 30 bacteria and is observed to double every 4 hours.

(a) Find an exponential model [tex]n(t) = n_0 \cdot 2^{t/T}[/tex] for the number of bacteria in the culture after [tex]t[/tex] hours.

(b) Estimate the number of bacteria after 17 hours. (Round your answer to the nearest whole number.)

(c) After how many hours will the bacteria count reach 1 million? (Round your answer to one decimal place.)

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Bird Population in a County

A certain species of bird was introduced in a certain county 25 years ago. Biologists observe that the population doubles every 10 years, and now the population is 13,000.

(a) What was the initial size of the bird population? (Round your answer to the nearest whole number.)

(b) Estimate the bird population 2 years from now. (Round your answer to the nearest whole number.)

(c) Sketch a graph of the bird population. (Assume [tex]t = 0[/tex] corresponds to the initial introduction).

To find an exponential model for the number of bacteria in the culture, use the formula n(t) = n0 * (2)^(t/k). After 17 hours, there will be approximately 503 bacteria. It will take about 60.2 hours for the bacteria count to reach 1 million.

To find an exponential model for the number of bacteria in the culture after t hours, we can use the formula n(t) = n0 * [tex](2)^(t/k)[/tex], where n(t) is the number of bacteria at time t, n0 is the initial number of bacteria, and k is the doubling time. In this case, n0 = 30 and k = 4. So, the exponential model for the number of bacteria is n(t) = 30 * (2)^(t/4).

To estimate the number of bacteria after 17 hours, we can substitute t = 17 into the exponential model. n(17) = 30 * [tex](2)^(17/4)[/tex] = 30 * 16.75 ≈ 502.5. Rounding to the nearest whole number, the estimated number of bacteria after 17 hours is 503 bacteria.

To find how many hours it will take for the bacteria count to reach 1 million, we can set n(t) = 1 million in the exponential model and solve for t. 1 million = 30 * [tex](2)^(t/4)[/tex]. Taking the logarithm of both sides, t/4 = log(base 2)(1 million/30). Solving for t, we get t ≈ 4 * log(base 2)(33333.33) ≈ 4 * 15.0584 ≈ 60.234. Rounding to one decimal place, it will take approximately 60.2 hours for the bacteria count to reach 1 million.

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