Question

The population of a southern city follows the exponential law. Use this information to answer parts a through b. (a) If N is the population of the city and t is the time in years, express N as a function of t. N(t)-No ekt (Type an expression using t as the variable and in terms of e.) (b) If the population doubled in size over 14 months and the current population is 10,000, what will the population be 4 years from now? The population will be approximately people. (Do not round until the final answer. Then round to the nearest whole number as needed.)

Answer 1

Answer:

• N(t) = N0·e^(kt)
• 107,672

Step-by-step explanation:

a) Your question supplies the formula for N(t):

  N(t) = N0·e^(kt)

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b) Given the initial population and doubling time (in months), the population function can also be written as ...

  N(t) = 10,000·2^(t/14) . . . . . t in months

Then in 4 years the population will be ...

  N(48) = 10,000·2^(48/14) ≈ 107,672.02

The population 4 years from now will be approximately 107,672 people.

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Comment on the formula for N(t)

In order to use the formula of the first part in answering the second part, we need to find the value of k. It will be an irrational number. In order to obtain accurate results in the second part, k would need to be good to at least 6 significant digits. Its value, for t in months, is ...

  k = ln(2)/14 ≈ 0.0495105

For t in years, it is 12 times this value, or ...

  k ≈ 0.594126

The advice not to do any rounding (even for the value of k) is appropriate.

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As we can see from the above, it is not necessary to determine k in order to answer the question in part b.