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 = 10yâ‚€, where yâ‚€ is the initial level of the population.
Let's solve the equation:
10yâ‚€ = 50e^(0.5t)
First, divide both sides of the equation by yâ‚€:
10 = 50e^(0.5t) / yâ‚€
Next, divide both sides of the equation by 50:
10/50 = e^(0.5t) / yâ‚€
Simplify the left side of the equation:
1/5 = e^(0.5t) / yâ‚€
Now, multiply both sides of the equation by yâ‚€:
yâ‚€/5 = e^(0.5t)
To isolate e^(0.5t), multiply both sides of the equation by 5:
5(yâ‚€/5) = 5e^(0.5t)
Simplify the left side:
yâ‚€ = 5e^(0.5t)
Now, divide both sides of the equation by 5:
yâ‚€/5 = e^(0.5t)
To eliminate the natural logarithm, take the natural logarithm of both sides:
ln(yâ‚€/5) = ln(e^(0.5t))
Simplify the right side:
ln(yâ‚€/5) = 0.5t
Finally, divide both sides of the equation by 0.5:
ln(yâ‚€/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(yâ‚€/5) / 0.5.