SAT Question 90bcaa61: Nonlinear Functions answer and explanation with Desmos solution

It's given that $t$ minutes after an initial observation, the number of bacteria in a population is $60,000{\left(2\right)}^{\frac{t}{410}}$. This expression consists of the initial number of bacteria, $60,000$, multiplied by the expression $2^{\frac{t}{410}}$. The time it takes for the number of bacteria to double is the increase in the value of $t$ that causes the expression $2^{\frac{t}{410}}$ to double. Since the base of the expression $2^{\frac{t}{410}}$ is $2$, the expression $2^{\frac{t}{410}}$ will double when the exponent increases by $1$. Since the exponent of the expression $2^{\frac{t}{410}}$ is $\frac{t}{410}$, the exponent will increase by $1$ when $t$ increases by $410$. Therefore the time, in minutes, it takes for the number of bacteria in the population to double is $410$.