Math

QuestionIn 2000, Mexico's population was 100 million, growing at 1.53%1.53 \% per year. When will it reach 109 million? Round to nearest year. a. 2006 b. 2008 c. 2005 d. 2007 (Select One)

Studdy Solution

STEP 1

Assumptions1. The initial population in the year2000 is100 million. . The population growth rate is1.53% per year.
3. The population growth is exponential, modeled by the function f(x)=100(1.0153)xf(x)=100(1.0153)^x.
4. We are looking for the year when the population reaches109 million.

STEP 2

First, we need to set up the equation to solve for the year when the population reaches109 million. We can do this by setting the function f(x)f(x) equal to109.
f(x)=109f(x) =109

STEP 3

Substitute the given function into the equation.
100(1.0153)x=109100(1.0153)^x =109

STEP 4

To solve for xx, we first need to isolate the exponential term. We can do this by dividing both sides of the equation by100.
1.0153x=1.091.0153^x =1.09

STEP 5

Next, we need to get rid of the base1.0153. We can do this by taking the natural logarithm (ln) of both sides of the equation.
ln(1.0153x)=ln(1.09)\ln(1.0153^x) = \ln(1.09)

STEP 6

Using the property of logarithms that allows us to bring down the exponent, we getxln(1.0153)=ln(1.09)x \ln(1.0153) = \ln(1.09)

STEP 7

Now, we can solve for xx by dividing both sides of the equation by ln(1.0153)\ln(1.0153).
x=ln(1.09)ln(1.0153)x = \frac{\ln(1.09)}{\ln(1.0153)}

STEP 8

Calculate the value of xx using a calculator.
x5.8x \approx5.8

STEP 9

Since xx represents the number of years after200, we need to add200 to xx to find the year when the population reaches109 million.
Year=200+xYear =200 + x

STEP 10

Substitute the value of xx into the equation.
Year=2000+5.8Year =2000 +5.8

STEP 11

Since we can't have a fraction of a year, we need to round xx to the nearest whole number. In this case, 5.85.8 rounds up to 66.
Year=2000+6Year =2000 +6

STEP 12

Calculate the year.
Year=2000+6=2006Year =2000 +6 =2006So, according to the model, the population of Mexico would reach109 million in the year2006.

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