How long will it take for the population to reach 10 times its initial level?
y=50(e)^(0.5t)
To find out how long it will take for the population to reach 10 times its initial level, we need to find the value of t that satisfies the equation y = 10y₀, where y₀ is the initial level of the population.
Let's solve the equation:
10y₀ = 50e^(0.5t)
First, divide both sides of the equation by y₀:
10 = 50e^(0.5t) / y₀
Next, divide both sides of the equation by 50:
10/50 = e^(0.5t) / y₀
Simplify the left side of the equation:
1/5 = e^(0.5t) / y₀
Now, multiply both sides of the equation by y₀:
y₀/5 = e^(0.5t)
To isolate e^(0.5t), multiply both sides of the equation by 5:
5(y₀/5) = 5e^(0.5t)
Simplify the left side:
y₀ = 5e^(0.5t)
Now, divide both sides of the equation by 5:
y₀/5 = e^(0.5t)
To eliminate the natural logarithm, take the natural logarithm of both sides:
ln(y₀/5) = ln(e^(0.5t))
Simplify the right side:
ln(y₀/5) = 0.5t
Finally, divide both sides of the equation by 0.5:
ln(y₀/5) / 0.5 = t
Therefore, the time it will take for the population to reach 10 times its initial level is given by t = ln(y₀/5) / 0.5.