v(t)= Ce^(k(square root(t))
Suppose that the dealer, who is 25 years old, decides to sell the card at time , sometime in the next 40 years: 0< or equal to t < or equal to 40. At that time , he’ll invest the money he gets for the sale of the card in a bank account that earns an interest rate of r , compounded continuously. (This means that after years, an initial investment of will be worth Ie^(rt).) When he turns 65, he’ll take the money that’s in his bank account and begin to draw on it for his retirement. Let A be the amount of money in his account when he turns 65.
6. If those values of the constants were accurate, then when should the dealer sell the card so as to maximize the amount in his retirement account when he turns 65?
To determine when the dealer should sell the card to maximize the amount in his retirement account when he turns 65, we need to find the time, denoted as t, that maximizes the function A(t) representing the amount of money in his account at age 65.
Given that v(t) represents the value of the card at time t, we can write the amount of money in the dealer's account at age 65 as:
A(t) = v(t)e^(r(65-t))
To maximize the amount A(t), we need to find the value of t that maximizes A(t).
First, let's rewrite the equation for v(t) using the given equation v(t) = Ce^(k√t):
A(t) = Ce^(k√t)e^(r(65-t))
Since e^x * e^y = e^(x+y), we can simplify the equation further:
A(t) = Ce^(k√t + r(65-t))
To maximize A(t), we need to find the critical points by taking the derivative of A(t) with respect to t and setting it equal to zero:
A'(t) = C(k/2√t)e^(k√t + r(65-t)) - Cr*e^(k√t + r(65-t)) = 0
Next, let's solve for t:
C(k/2√t)e^(k√t + r(65-t)) = Cr*e^(k√t + r(65-t))
Dividing both sides by C and canceling out the common factors:
(k/2√t)e^(k√t + r(65-t)) = re^(k√t + r(65-t))
Now, we can simplify the equation:
(k/2√t) = r
To solve for t, we can square both sides:
k^2/(4t) = r^2
Solving for t, we get:
t = k^2/(4r^2)
Therefore, the dealer should sell the card when t equals k^2/(4r^2) to maximize the amount in his retirement account when he turns 65.