A seller in ebay has a rare basketball, which was used in 2012 London Olympics and...

  

 

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A seller in ebay has a rare basketball, which was used in 2012 London Olympics and was signed by the basketball giant, Kobe Bryant. The seller is trying to decide when to sell it, and he is asking for 980 USD at this stage. He knows its value will grow over time, but he could sell it and invest the money in a bank account, and the value of the money would also grow over time due to interest. The question is: when should the seller sell the basketball? Experience suggests to the seller that over time, the value of the basketball, like many other collectibles, will grow in a way consistent with the following model: V(t) = Aeºvt where A and are constants, and V(t) is the value of the basketball in dollars at the time t years after the present time. . 1. Plot this function against t when A = . 2. What is the interpretation of A? 980 and 0 0.5. . 3. Plot the function V (t) for several different values of 0. What effect does have on the value of the basketball over time? Suppose that the seller, who is 35 years old, decides to sell this basketball at time t, sometime in the next 30 years: 0 ≤ t ≤ 30. At that time t, he will invest the money he gets from the sale in a bank account that earns an interest rate of r, compounded continuously, which means that after t years, an initial investment of B USD will be worth Bert USD. When he turns 65, he will take the money in his bank account for his retirement. Let M(t) be the amount of money in his account when he turns 65, where t is the time at which he sells his basketball. 4. Write down the closed-formed expression of M (t). • 5. Plot your function M(t) against t when A= 980, 0 = 0.5, and r = 0.05. . 6. If those values of the constants were accurate, then when should the seller sell the basket- ball to maximize the amount in his retirement account when he turns 65? . 7. Plot the function M(t) for several different values of 0, while holding r constant. What does a larger value of 0 imply about the value of the basketball over time? (Refer back to question 3.) And now, what does a larger value of imply about the best time to sell the basketball? Do these two facts seem consistent with one another? . 8. Plot the function M(t) for several different values of r, while holding constant. What does a larger value of r imply about the best time to sell the basketball? Is that consistent with the meaning of r? . 9. Let to be the optimal time to sell the basketball, i.e., the time that will maximize M (t). Try to find to in the general model. Note that your solution should be a function of the constant variables A, 0 and r. . 10. Plot M (t) against t for different combinations of A, 0 and r, and verify that your expres- sion for to does accurately predict when the best time will be to sell the basketball. . 11. Are the properties of to as it relates to and r consistent with what you found in step 7 and step 8? 12. There is another way to decide when to sell the basketball instead of thinking about putting the money from the sale into a retirement account. Suppose that today (time=0) the seller puts an amount of money W 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 Wert. If at time t the basketball is sold for an amount equal to V (t) = Aevt, how much money W would the seller 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 W is called the present value of the basketball if it ends up being sold at time t. Model the present value of the basketball as a function of the time t 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? For the amount to equal, V(t) = Ae^(esqrt(t)) = We^(rt). So W = (Ae^esqrt(t)) / (e^(rt)) = Ae^(t(1/0-r)). The time when selling the card would maximize its present value is DW/DT= (1/0 - r) * A *e^(t(1/0 - r)), maximized at DW/DT= 0. Yes, the answers are consistent with the ones we found earlier in step 9. A seller in ebay has a rare basketball, which was used in 2012 London Olympics and was signed by the basketball giant, Kobe Bryant. The seller is trying to decide when to sell it, and he is asking for 980 USD at this stage. He knows its value will grow over time, but he could sell it and invest the money in a bank account, and the value of the money would also grow over time due to interest. The question is: when should the seller sell the basketball? Experience suggests to the seller that over time, the value of the basketball, like many other collectibles, will grow in a way consistent with the following model: V(t) = Aeºvt where A and are constants, and V(t) is the value of the basketball in dollars at the time t years after the present time. . 1. Plot this function against t when A = . 2. What is the interpretation of A? 980 and 0 0.5. . 3. Plot the function V (t) for several different values of 0. What effect does have on the value of the basketball over time? Suppose that the seller, who is 35 years old, decides to sell this basketball at time t, sometime in the next 30 years: 0 ≤ t ≤ 30. At that time t, he will invest the money he gets from the sale in a bank account that earns an interest rate of r, compounded continuously, which means that after t years, an initial investment of B USD will be worth Bert USD. When he turns 65, he will take the money in his bank account for his retirement. Let M(t) be the amount of money in his account when he turns 65, where t is the time at which he sells his basketball. 4. Write down the closed-formed expression of M (t). • 5. Plot your function M(t) against t when A= 980, 0 = 0.5, and r = 0.05. . 6. If those values of the constants were accurate, then when should the seller sell the basket- ball to maximize the amount in his retirement account when he turns 65? . 7. Plot the function M(t) for several different values of 0, while holding r constant. What does a larger value of 0 imply about the value of the basketball over time? (Refer back to question 3.) And now, what does a larger value of imply about the best time to sell the basketball? Do these two facts seem consistent with one another? . 8. Plot the function M(t) for several different values of r, while holding constant. What does a larger value of r imply about the best time to sell the basketball? Is that consistent with the meaning of r? . 9. Let to be the optimal time to sell the basketball, i.e., the time that will maximize M (t). Try to find to in the general model. Note that your solution should be a function of the constant variables A, 0 and r. . 10. Plot M (t) against t for different combinations of A, 0 and r, and verify that your expres- sion for to does accurately predict when the best time will be to sell the basketball. . 11. Are the properties of to as it relates to and r consistent with what you found in step 7 and step 8? 12. There is another way to decide when to sell the basketball instead of thinking about putting the money from the sale into a retirement account. Suppose that today (time=0) the seller puts an amount of money W 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 Wert. If at time t the basketball is sold for an amount equal to V (t) = Aevt, how much money W would the seller 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 W is called the present value of the basketball if it ends up being sold at time t. Model the present value of the basketball as a function of the time t 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? For the amount to equal, V(t) = Ae^(esqrt(t)) = We^(rt). So W = (Ae^esqrt(t)) / (e^(rt)) = Ae^(t(1/0-r)). The time when selling the card would maximize its present value is DW/DT= (1/0 - r) * A *e^(t(1/0 - r)), maximized at DW/DT= 0. Yes, the answers are consistent with the ones we found earlier in step 9.

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