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.
4. Write an expression for A(t), the amount of money in the dealer’s account when he turns 65 written as a function of t , the time at which he sells the baseball card.
money from card sale at time t:
v(t) = Ce√(kt)
time left to draw interest: 40-t
A = Ce√(kt) * e^(r(40-t))
= Ce^(√(kt)+r(40-t))
To find the expression for A(t), the amount of money in the dealer's account when he turns 65, we need to consider the following:
1. The initial investment: This is the money he receives for selling the baseball card.
2. The interest rate: The bank account earns an interest rate of r, compounded continuously. This means that after t years, an initial investment I will be worth Ie^(rt).
3. The time at which he sells the baseball card: Let's denote this time as t.
4. The time when he turns 65: This time should be 65 - 25 = 40 years after he sells the card.
Now let's put all this information together to find the expression for A(t):
Step 1: Calculate the initial investment:
The dealer sells the card at time t. The initial investment is the money he receives for selling the card, which we'll denote as C. So, the initial investment is C.
Step 2: Calculate the time from selling the card to turning 65:
The dealer sells the card at time t, and he turns 65 after 40 years. So, the time from selling the card to turning 65 is 40 - t years.
Step 3: Calculate the amount of money in the account when he turns 65:
According to the formula for continuous compound interest, the amount of money in the account after (40 - t) years will be:
A(40 - t) = Ce^(r(40 - t))
Therefore, the expression for A(t), the amount of money in the dealer's account when he turns 65, written as a function of t, is:
A(t) = Ce^(r(40 - t))