I have a table and was given an equation of best fit - the table is about the growing US population. My equation of best fit is y = 106.7236(1.0124)^x
I found my y intercept and the population in 2013. Now it is asking when will the US population break 400 million. In 2013 the population is 335.7 million. Is there an easier way to solve this, or do I just have to do trial and error?
Because I'm not sure how to solve the equation 400 = 106.7236(1.0124)^x?????
log(400/106.7236) = x log(1.0124)
I'm sorry, but I really have no idea what this means....- what is log?
never heard of logarithms?
there is a log key on calculators
No, I am in basic math
To determine when the US population will reach 400 million, you can solve the equation y = 106.7236(1.0124)^x for x when y is 400. While trial and error is one possible approach, there is a more efficient way to solve this type of exponential equation.
The given equation represents exponential growth, where y represents the population at a given year (x). Let's solve the equation step by step to find when the US population will break 400 million:
1. Start with the equation: y = 106.7236(1.0124)^x
2. Replace y with 400: 400 = 106.7236(1.0124)^x
3. Divide both sides of the equation by 106.7236: 400/106.7236 = (1.0124)^x
4. Simplify the left side of the equation: 3.747893 = (1.0124)^x
Now, to solve for x, you can take the logarithm of both sides of the equation. The logarithm function is the inverse of exponential function, which allows you to isolate the exponent. The most common choice for logarithm is the natural logarithm (ln), but you can also use other logarithmic bases:
5. Take the logarithm of both sides: ln(3.747893) = ln((1.0124)^x)
6. Apply the logarithmic property: ln(3.747893) = x ln(1.0124)
7. Divide both sides of the equation by ln(1.0124): ln(3.747893) / ln(1.0124) = x
Now, calculate the value of x using a calculator:
8. Use a calculator to find: x ≈ 82.51
Rounding x to the nearest whole number, you can conclude that the US population will break 400 million about 83 years after 2013. To find the specific year, add x to the starting year of 2013:
9. Calculate the year: 2013 + 83 = 2096
Therefore, if the given equation accurately represents the US population growth, the US population will likely break 400 million around the year 2096.