The population of a city can be modeled using the formula P=100,000e^(0.05t), where t is the number of years after 2012 and P is the city's population. Which of the following equations can be used to find the number of years after 2012 that the population will triple to 300,000?
t=ln3/.05
P(t) = 300,000
The population of a city can be modeled using the formula , where t is the number of years after and P is the city's population. Which of the following equations can be used to find the number of years after that the population will triple to ?
Ah, calculations! Brace yourself for some mathematical hilarity. To find the number of years after 2012 that the population will triple to 300,000, we need to set up an equation. So, let's plug in the numbers we have. We want the population (P) to be 300,000, which means our equation will be 300,000 = 100,000e^(0.05t). Now, to solve this equation, we'll need to use some algebraic acrobatics. We need to isolate the variable "t," so let's divide both sides of the equation by 100,000 first. And there you have it! Your equation to find the number of years after 2012 that the population will triple to 300,000 is: e^(0.05t) = 3. Now go forth and solve with an added touch of humor, my friend!
To find the number of years after 2012 that the population will triple to 300,000, we need to set up an equation where we solve for the value of t.
Given that the population formula is P = 100,000e^(0.05t), we can set up an equation as follows:
300,000 = 100,000e^(0.05t)
To solve this equation for t, we need to isolate the variable t. Let's go through the steps:
Step 1: Divide both sides of the equation by 100,000 to simplify:
300,000 / 100,000 = e^(0.05t)
Step 2: Simplify the division:
3 = e^(0.05t)
Now, the equation is simplified to 3 = e^(0.05t).
To solve for t, we need to take the natural logarithm (ln) of both sides of the equation:
ln(3) = ln(e^(0.05t))
Using the property of logarithms that ln(a^b) = bln(a), we can rewrite the equation as:
ln(3) = 0.05t * ln(e)
Since ln(e) is equal to 1, we can further simplify the equation to:
ln(3) = 0.05t
Finally, to isolate t, we divide both sides of the equation by 0.05:
ln(3) / 0.05 = t
Now, we have the value of t that represents the number of years after 2012 that the population will triple to 300,000.
Therefore, the correct equation is t = ln(3) / 0.05.