during one time period, the price of rhodium increased at a rate that was proportional to the price of…

Explanation:

Step1: Identify the exponential - growth formula

The general formula for exponential growth is $P(t)=P_0\cdot a^{\frac{t}{k}}$, where $P_0$ is the initial amount, $a$ is the growth factor, $t$ is the time elapsed, and $k$ is the time it takes for the quantity to change by a factor of $a$. Here, $P_0 = 475$, $a = 4$ (since it quadruples), and $k = 25$. So the formula for the price of rhodium as a function of time $t$ (in months) is $P(t)=475\cdot4^{\frac{t}{25}}$.

Step2: Substitute $t = 18$ into the formula

We want to find $P(18)$. Substitute $t = 18$ into $P(t)=475\cdot4^{\frac{t}{25}}$. First, calculate the exponent: $\frac{18}{25}=0.72$. Then, find $4^{0.72}$. Using a calculator, $4^{0.72}\approx2.714$. Now, $P(18)=475\times2.714$.

Step3: Calculate the final price

$P(18)=475\times2.714 = 1289.15\approx1289$.

Answer:

C. $1289$