a. The function that models the situation is F(t) = 48(1.08)ˣ
b. The price of the stock 6 years from now is $76.17
c. Find the graph in the attachment
d. I receive a dividend of $0.026.
<h3>a. Write a function f(t) that models the situation.</h3>
Since the stock price increases at a rate of 8% every year, and is initially $48, it follows exponential growth.
So, the current price
where
- A = price at current moment = $48,
- r = rate of growth = 8% = 0.08 and
- t = number of years
So, ![F(t) = A(1 + r)^{t}](https://tex.z-dn.net/?f=F%28t%29%20%3D%20A%281%20%2B%20r%29%5E%7Bt%7D)
![F(t) = 48(1 + 0.08)^{t} \\= 48(1.08)^{t}](https://tex.z-dn.net/?f=F%28t%29%20%3D%2048%281%20%2B%200.08%29%5E%7Bt%7D%20%5C%5C%3D%2048%281.08%29%5E%7Bt%7D)
So, the function that models the situation is F(t) = 48(1.08)ˣ
<h3>b. Determine the price of the stock 6 years from now?</h3>
The price of the stock 6 years from now is gotten when t = 6.
So,
![F(t) = 48(1.08)^{t} \\= 48(1.08)^{6} \\= 48(1.5869)\\= 76.17](https://tex.z-dn.net/?f=F%28t%29%20%3D%2048%281.08%29%5E%7Bt%7D%20%5C%5C%3D%2048%281.08%29%5E%7B6%7D%20%5C%5C%3D%2048%281.5869%29%5C%5C%3D%2076.17)
So, the price of the stock 6 years from now is $76.17
<h3>c. Sketch a graph of the price of the function vs time in years</h3>
Find the graph in the attachment
<h3>d. Bonus</h3>
Since every quarter, the company pays a dividend of 1.5 %, the rate per year would be r = 1.5 % ÷ 1/4 year = 1.5 % × 4 = 6 % per year.
Since they pay at a rate, r = 6 % = 0.06 of the stock price, F(t) as dividend.
After n years, the dividend is D = (r)ⁿF(t)
= (0.06)ⁿF(t)
So, ![D = (0.06)^{t}F(t) \\= (0.06)^{t}[4.8(1.08)^{t}]](https://tex.z-dn.net/?f=D%20%3D%20%280.06%29%5E%7Bt%7DF%28t%29%20%5C%5C%3D%20%20%280.06%29%5E%7Bt%7D%5B4.8%281.08%29%5E%7Bt%7D%5D)
So, after 3 years when t = 3,
![D = (0.06)^{t}[4.8(1.08)^{t}]\\D = (0.06)^{3}[4.8(1.08)^{3}]\\D = 0.000216 \times 48 \times 1.2597\\D = 0.013](https://tex.z-dn.net/?f=D%20%3D%20%280.06%29%5E%7Bt%7D%5B4.8%281.08%29%5E%7Bt%7D%5D%5C%5CD%20%3D%20%280.06%29%5E%7B3%7D%5B4.8%281.08%29%5E%7B3%7D%5D%5C%5CD%20%3D%200.000216%20%5Ctimes%2048%20%5Ctimes%201.2597%5C%5CD%20%3D%200.013)
Since there are 3 shares, the total dividend would be D' = 3D
= 3 × 0.013
= 0.026
So, i receive a dividend of $0.026
Learn more about exponential function here:
brainly.com/question/12940982
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