81. STOCK The price per share of a coffee chain's stock was \( \$ 0.93 \) in a month during its first year of trading. During its fifth year of trading, the price per share of stock was \( \$ 3.52 \) during the same month. (Example 9) a. Write a continuous exponential equation to model the price of stock \( P \) as a function of year of trading \( t \). Round \( k \) to the nearest ten-thousandth. b. Use the model you found in part a to predict the price of the stock during the ninth year of trading.

To create a continuous exponential equation for the stock price \( P \) as a function of the year of trading \( t \), we start with the general formula: \[ P(t) = P_0 \cdot e^{kt} \] Where \( P_0 \) is the initial price, \( k \) is the growth rate, and \( t \) is the number of years since trading began. ### a. Finding \( P_0 \) and \( k \) Given: - In the first year (\( t=1 \)), \( P(1) = 0.93 \) - In the fifth year (\( t=5 \)), \( P(5) = 3.52 \) From \( P(1) \): \[ P(1) = P_0 \cdot e^{k \cdot 1} = P_0 e^{k} = 0.93 \] From \( P(5) \): \[ P(5) = P_0 \cdot e^{k \cdot 5} = P_0 e^{5k} = 3.52 \] Now we can divide the equations to solve for \( k \): \[ \frac{P_0 e^{5k}}{P_0 e^{k}} = \frac{3.52}{0.93} \] This simplifies to: \[ e^{4k} = \frac{3.52}{0.93} \approx 3.7828 \] Taking the natural logarithm of both sides: \[ 4k = \ln(3.7828) \] \[ k \approx \frac{\ln(3.7828)}{4} \approx 0.3355 \] Now substituting \( k \) back to find \( P_0 \): \[ 0.93 = P_0 e^{0.3355} \] \[ P_0 \approx \frac{0.93}{1.3993} \approx 0.6653 \] So the continuous exponential equation is: \[ P(t) = 0.6653 \cdot e^{0.3355t} \] ### b. Predicting the stock price in the ninth year Now, to find the predicted price of stock during the ninth year of trading (\( t=9 \)): \[ P(9) = 0.6653 \cdot e^{0.3355 \cdot 9} \] \[ P(9) \approx 0.6653 \cdot e^{3.0195} \] \[ P(9) \approx 0.6653 \cdot 20.3264 \approx 13.5253 \] Thus, the predicted price of the stock during the ninth year of trading is approximately **$13.53**. And there you have it! A dive into the ups and downs of stock pricing with a sprinkle of math magic!