A coin sold for $252 in 1980 and was sold again in 1987 for $439. Assume that th

Question

A coin sold for $252 in 1980 and was sold again in 1987 for $439. Assume that the growth in the value V of the collector's item was exponential. a) Find the value k of the exponential growth rate. Assume Vo252. (Round to the nearest thousandth.) b) Find the exponential growth function in terms of t, where t is the number of years since 1980. c) Estimate the value of the coin in 2010. (Round to the nearest dollar.) d) What is the doubling time for the value of the coin to the nearest tenth of a year? years (Round to the nearest tenth.)

Explanation / Answer

a. Let the exponential model be V(t)= V0 ekt, where V0 is the value of the coin in 1980 , V(t) is the value t years from 1980, and k is the constant of exponential growth. Here, V0 = 252 and in 1987, when t = 7, we have V(t) =439.Then , 439 = 252e7k or, e7k = 439/252. Now, on taking natural logarithm of both the sides, we get ln e7k = ln(439/252) or, 7k ln e = ln 439-ln 252 ( as log ab = blog a and log (m/n) = log m –log n). Thus, 7k = 6.084499413-5.529429088 ( as ln e = 1) or, 7k = 0.555070325 so that k = 0.555070325/7 =0.07929576 = 0.079( on rounding off to the nearest thousandth).

b. In view of part a) above, we have V(t)= 252 e0.079t.

c. In 2010, t = 30 so that the value of the coin in 2010 is 252e0.079*30 = 252 e2.37 = 252 *10.69739228 = $ 2696 ( on rounding off to the nearest dollar).

d. Let the doubling time for the value of the coin be t years. Then 2*252=252e0.079t or, e0.079t =2. Now, on taking natural logarithm of both the sides, we get ln e0.079 t = ln 2 or, 0.079t = 0.69314718 so that t =0.69314718 /0.079 = 8.774014944 = 8.8 years ( on rounding off to the nearest tenth of an year).

e. The amount is unclear so we make a presumption. Let the time for the value of the coin to reach $ 2981 be t years. Then 2981 =252e0.079t or, or, e0.079t =2981/252. Now, on taking natural logarithm of both the sides, we get 0.079t = ln 2981-ln 252 or, 0.079t = 8-5.529 =2.471 so that t =2.471 /0.079 = 31.3 years (on rounding off to the nearest tenth of an year).

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.