Question

The population of South Carolina can be approximated using the formula y = 1.264e⁰.⁰¹¹ˣ, where x is the number of years since 1900 and y is the population in millions. Use this equation to estimate the year that the population will reach 5.0 × 10⁶ people.

Answer 1

To estimate the year in which the population will reach 5.0 × 10⁶ people, we solve the exponential equation for x and approximate the number of years.

Step-by-step explanation:

To estimate the year in which the population will reach 5.0 × 10⁶ people, we need to solve the equation y = 1.264e⁰.⁰¹¹ˣ for x. Let's substitute y with 5.0 × 10⁶ and solve for x:

5.0 × 10⁶ = 1.264e⁰.⁰¹¹ˣ

To isolate e⁰.⁰¹¹ˣ, divide both sides of the equation by 1.264:

5.0 × 10⁶ / 1.264 = e⁰.⁰¹¹ˣ

Simplify the left side of the equation:

3.952 × 10⁶ ≈ e⁰.⁰¹¹ˣ

To solve for x, take the natural logarithm of both sides of the equation:

ln(3.952 × 10⁶) = ln(e⁰.⁰¹¹ˣ)

Use the property of logarithms to bring down the exponent:

ln(3.952 × 10⁶) = 0.011x

Divide both sides of the equation by 0.011:

x ≈ ln(3.952 × 10⁶) / 0.011

Use a calculator to compute the right side of the equation:

x ≈ 62.204

Rounded to the nearest whole number, the population will reach 5.0 × 10⁶ people approximately 62 years after 1900.