calculus

posted by .

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?

Similar Questions

1. Economics/Math

3. Assume you own a painting. If you sold it now, you could get £500 for it. However, the amount that people will be willing to pay in the future increases by £25 in odd years (including the first one) and by £26 in even years, …
2. math

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 …
3. calculus

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 …
4. math

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 …
5. Calculus

square root of (12), square root of (12 + square root of (12)), square root of (12+ square root of (12+ square root of (12))) Write a recursion formaula for an(subcription), n greater than or equal to 2.
6. Statistic

A Toyota dealer wants to find out the age of their customers (for advertising purposes). They want the margin of error to be 3 years old. Suppose the sigma is 13 years old. If they want a 90% confidence interval, how many people do …
7. Algebra

Let x equal the age, in human years, for a dog that is 2 years old or older. For each of your chosen breeds, write an expression to model its age in dog years. [Hint: Use (x - 2) in each expression.] These are the dog years: 2yrs: …
8. Math

Kathy is y years old Find her age,if 1/2 of wat she was 3 years ago is equal to 1/3 of what she will be in 4 years time
9. Math

Let x equal the age, in human years, for a dog that is 2 years old or older. For each of your chosen breeds, write an expression to model its age in dog years. [Hint: Use (x-2) in each expression.] the dogs are 2yrs: 23 yrs old 3yrs: …
10. Geometry

Let x equal the age, in human years, for a dog that is 2 years old or older. For each of your chosen breeds, write an expression to model its age in dog years. [Hint: Use (x-2) in each expression.] the dogs are 2yrs: 23 yrs old 3yrs: …

More Similar Questions