How long will it take for the population to reach 10 times its initial level?

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.