Answer
$3$ years.
Work Step by Step
Our goal is to determine $t$ so that $f(t)=125$.
Replace $f(t)$ with $125$ and solve the following equation for $t$:
$\Rightarrow 125=\frac{250(3t+5)}{t+25}$
Multiply both sides by $t+25$ to clear fractions.
$\Rightarrow 125(t+25)=(t+25)\left (\frac{250(3t+5)}{t+25}\right )$
Use the distributive property.
$\Rightarrow 125t+3125=750t+1250$
Add $-125t-1250$ to both sides.
$\Rightarrow 125t+3125-125t-1250=750t+1250-125t-1250$
Simplify.
$\Rightarrow 1875=625t$
Divide both sides by $625$.
$\Rightarrow \frac{1875}{625}=\frac{625t}{625}$
Simplify.
$\Rightarrow 3=t$
Hence, it will take $3$ years for the population to increase to $125$ elk.
