Question

a vineyard planted 50 grapevines in 2018. by 2021, the number of grapevines had increased to 400 due to ideal growing conditions. assuming exponential growth, express the population of grapevines as a function of time. select one: a. $n(t) = 50(2)^t$ b. $n(t) = 50(1.6818)^t$ c. $n(t) = 50(0.5)^t$ d. $n(t) = \frac{350}{3}t + 50$

Explanation:

Step1: Recall exponential growth formula

The general exponential growth function is $N(t) = N_0(r)^t$, where $N_0$ is the initial population, $r$ is the growth factor, and $t$ is time in years.

Step2: Define known values

Initial year 2018: $N_0 = 50$, $t=0$. Year 2021: $t=2021-2018=3$, $N(3)=400$.

Step3: Solve for growth factor $r$

Substitute values into the formula:
$$400 = 50(r)^3$$
Divide both sides by 50:
$$\frac{400}{50} = r^3$$
$$8 = r^3$$
Take the cube root of both sides:
$$r = \sqrt[3]{8} = 2$$

Step4: Form the final function

Substitute $N_0=50$ and $r=2$ into the exponential growth formula.

Answer:

A. $N(t) = 50(2)^t$