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In the year 2010, the population of a city was 22
million and was growing at a rate of about 2.3%
per year. The function p(t) = 22(1.023)' gives the
population, in millions, 1 years after 2010. Use
the model to determine in what year the
population will reach 30 million. Round to the
nearest year.

Respuesta :

maymayisadayday123

Answer:

1) The function p(t) = 22(1.023)ᵗ models the population (in millions) t years after 2010.

2) We want to find the value of t when the population reaches 30 million. In other words, we need to solve the equation:

  p(t) = 30

3) Substituting the expression for p(t), we get:

  22(1.023)ᵗ = 30

4) To solve this, let's take the logarithm of both sides. Since the base of the exponential is 1.023, we'll use the logarithm to base 1.023:

  log₁.₀₂₃(22(1.023)ᵗ) = log₁.₀₂₃(30)

5) Using the logarithm property log(ab) = log(a) + log(b), we can separate the left side:

  log₁.₀₂₃(22) + log₁.₀₂₃((1.023)ᵗ) = log₁.₀₂₃(30)

6) Using the logarithm property log(aᵇ) = b · log(a), we can simplify the second term on the left:

  log₁.₀₂₃(22) + t · log₁.₀₂₃(1.023) = log₁.₀₂₃(30)

7) Since log₁.₀₂₃(1.023) = 1, this simplifies to:

  log₁.₀₂₃(22) + t = log₁.₀₂₃(30)

8) Now we can solve for t:

  t = log₁.₀₂₃(30) - log₁.₀₂₃(22)

9) We can calculate this using a calculator or computer (many have a 'log' or 'ln' function where you can specify the base). Doing so, we get:

  t ≈ 13.7

10) Since we need to round to the nearest year, the population will reach 30 million in around 14 years after 2010, which is the year 2024.

Step-by-step explanation:

msm555

Answer:

Year: 2024

Step-by-step explanation:

To determine in what year the population will reach 30 million using the given model:

[tex]\sf p(t) = 22(1.023)^t[/tex]

where

[tex]\sf t[/tex] represents the number of years after 2010, we need to solve the equation:

[tex]\sf p(t) = 30[/tex]

Substituting [tex]\sf p(t)[/tex] with the given expression:

[tex]\sf 22(1.023)^t = 30[/tex]

Now, we need to solve for [tex]\sf t[/tex]. To do this, we can use the property of logarithms to isolate the exponent [tex]\sf t[/tex].

Taking the natural logarithm (ln) of both sides of the equation:

[tex]\sf \ln(22(1.023)^t) = \ln(30) [/tex]

Using the logarithmic property:

[tex]\sf \ln(ab) = \ln(a) + \ln(b)[/tex]:

We get

[tex]\sf \ln(22) + \ln((1.023)^t) = \ln(30) [/tex]

Using the logarithmic property:

[tex]\sf \ln(a^b) = b \cdot \ln(a)[/tex]:

We get;

[tex]\sf \ln(22) + t \cdot \ln(1.023) = \ln(30) [/tex]

Now, we isolate [tex]\sf t[/tex] by subtracting [tex]\sf \ln(22)[/tex] and dividing by [tex]\sf \ln(1.023)[/tex]:

[tex]\sf t = \dfrac{\ln(30) - \ln(22)}{\ln(1.023)} [/tex]

Using a calculator, we can find:

[tex]\sf t \approx \dfrac{3.4011973816621 - 3.0910424533583}{0.0227394869694} [/tex]

[tex]\sf t \approx \dfrac{0.3101549283038}{0.0227394869694} [/tex]

[tex]\sf t \approx 13.639486621661 [/tex]

[tex]\sf t \approx 14 \textsf{(in nearest year)} [/tex]

So, the population will reach 30 million approximately 14 years after 2010 or , which would be in the year 2024.

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