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? First estimate the answer using the graph, then use calculus to verify your answer.

7. Plot the function A(t) for several different values of k, while holding r constant. What does a larger value of k imply about the value of the card over time?. And now, what does a larger value of k imply about the best time to sell the card? Do these two facts seem consistent with one another?

8. Plot the function A(t) for several different values of r ,while holding k constant. What does a larger value of r imply about the best time to sell the card? Is that consistent with the meaning of r ?

9. Let t be the optimal time to sell the baseball card—i.e., the time that will maximize A(t). Use calculus to find t in the general model. Note that since the constants C,k,and r are part of the general model, they may be part of the solution as well.

10. Graph A(t) against t for different combinations of C,k,and, r and verify that
your expression for t does accurately predict when the best time will be to sell the baseball card.

11. Are the properties of t as it relates to k and r consistent with those you found in steps 7 and 8 above?

12. There is another way to decide when to sell the baseball card instead of thinking about putting the money from the sale into a retirement account. Suppose that today (time =0) the dealer puts an amount of money $M into a bank account that earns interest at an annual rate of r , compounded continuously, so that at time t the bank account will be worth $Me^(rt). If at time t the baseball card is sold for an amount
equal to V(t)=Ce^(k(sqrt(t)), how much money $M would the dealer have needed to invest initially in order for the bank account value and the baseball card value to be equal at the time of the sale? That amount $M is called the present value of the baseball card if it ends up being sold at time t.

Model the present value of the card as a function of the time when it is sold. Find the time when selling the card would maximize its present value. Is the answer consistent with the one you found earlier, in step 9 above?

To answer these questions, we need to understand the given function and use calculus concepts.

The function v(t) represents the value of the baseball card at time t. It is given as v(t) = Ce^(k√t), where C, k, and t are constants. The dealer wants to sell the card to maximize the amount in his retirement account when he turns 65.

6. To find the optimal time to sell the card, we need to maximize the amount in the retirement account. This amount can be represented as A(t) = Ie^(rt), where I is the initial investment and r is the interest rate.
First, estimate the answer using the graph by plotting v(t) and A(t) on the same graph. Look for the time when the value of v(t) intersects or reaches its maximum point with A(t).
Then, use calculus to verify the answer. Take the derivative of A(t) with respect to t, set it equal to zero, and solve for t. The value of t obtained will be the optimal time to sell the card.

7. Plot the function A(t) for several different values of k while keeping r constant. A larger value of k implies that the value of the card increases rapidly over time. This suggests that the dealer should sell the card earlier to maximize the amount in his retirement account when he turns 65. So, larger values of k indicate an earlier optimal time to sell the card.

8. Plot the function A(t) for several different values of r while keeping k constant. A larger value of r implies a higher interest rate, which means the money in the bank account grows faster. This suggests that the dealer should sell the card later to take advantage of the higher interest rate. Therefore, as r increases, the optimal time to sell the card will be later. This is consistent with the meaning of r as the interest rate.

9. To find the optimal time to sell the card using calculus, we need to maximize the function A(t). Take the derivative of A(t) with respect to t, set it equal to zero, and solve for t. This will give us the value of t that maximizes A(t). Note that C, k, and r are part of the general model, so they may be included in the solution.

10. Graph A(t) against t for different combinations of C, k, and r, and verify that the expression for t obtained in step 9 accurately predicts the best time to sell the card. Compare the plotted graph with the estimated answer from step 6.

11. Compare the properties of t obtained in step 9 with those found in steps 7 and 8. Check if they are consistent with each other. For example, if in step 7, a larger value of k implied an earlier optimal time to sell the card, and in step 8, a larger value of r implied a later optimal time to sell, see if the t values obtained in step 9 align with these observations.

12. Consider the present value of the card when it is sold at time t. The present value is the amount that the dealer would have needed to initially invest in a bank account with interest rate r to have an equal value as the baseball card at the time of sale.
Model the present value as a function of time t. Find the time when selling the card would maximize its present value. Compare this answer with the one obtained in step 9 to check if they are consistent with each other.

6. To find the optimal time for the dealer to sell the card and maximize the amount in his retirement account, we can first estimate the answer using the graph of the function A(t). This graph represents the amount of money in his account when he turns 65.

7. Plotting the function A(t) for different values of k while holding r constant allows us to understand the effect of the constant k on the value of the card over time. A larger value of k implies that the card's value will increase at a higher rate over time. With a higher k, the best time to sell the card would be earlier, as the value increases rapidly.

8. Similarly, plotting the function A(t) for different values of r while holding k constant helps us understand the effect of the interest rate r on the best time to sell the card. A larger value of r implies a higher growth rate for the retirement account, so the best time to sell the card would be later when the account has had more time to grow.

9. To find the optimal time to sell the card, t, we can use calculus to maximize the function A(t). Taking the derivative of A(t) with respect to t and setting it equal to zero, we can solve for t. The constants C, k, and r will be part of the solution as well.

10. By graphing A(t) for different combinations of C, k, and r, we can verify if our expression for t accurately predicts the best time to sell the card.

11. The properties of t as it relates to k and r should be consistent with the observations made in steps 7 and 8 above. A larger value of k should result in an earlier optimal time to sell the card, and a larger value of r should result in a later optimal time to sell.

12. Another way to decide when to sell the card is to compare its value to the value of a bank account that earns interest at rate r. If the present value of the card (the initial investment needed to make its value equal to the bank account value at the time of sale) is calculated as a function of time, we can find the time that maximizes this present value. We can then compare this answer to the optimal time calculated earlier in step 9 to check for consistency.