Test: Test 3 Su This Question: 1 pt 8 of 4D (21 complete) This Test: 40 pts A do
ID: 3184977 • Letter: T
Question
Test: Test 3 Su This Question: 1 pt 8 of 4D (21 complete) This Test: 40 pts A doll sold for $245 in 1975 and was sold again in 1969 for $412. Assume that the growth in the value V of the collector's idem was exponential a) Find the value k of the exponential growth rate. Assume Vo-245 Round to the nearest thousandth) b) Find the exponential growth function in terms of t where t is the number of years since 1975 c) Estimate the value of the doll in 2011 Round to the nearest dollar) d) What is the doubling time for the value of the doll to the nearost tenth of a year? years Round to the nearest tenth ) e) Find the amount of time after which the value of the doll wall be $1327 Round to the nearest tenth) Enter your answer in each of the answer boxesExplanation / Answer
a). Let the exponential growth function determining the future value of the doll be V(t) = V0ekt where t is the number of years from 1975 , V0 is its initial value in 1975 and V(t) is its value t years from 1975.
Since the value of the doll was $ 245 in 1975, hence V0 = 245 so that V(t) = 245ekt. Further, the value of the doll was $ 412 in 1989, when t = 14 so that 412 = 245e14k or, e14k = 412/245.Now, on taking natural log of both the sides, we get 14kln e=ln 412-ln 245 or, 14k = 6.021023349-5.501258211 = 0.519765138. Hence, k = 0.519765138/14 = 0.037126081= 0.037 ( on rounding off to the nearest thousandth).
b). The exponential growth function determining the future value of the doll is V(t) = 245e0.037t where t ande V(t) are as above.
c).In 2011, t = 36 so that the value of the doll in 2011 will be 245 e0.037*36 =245e1.332 = 245*3.788613041 =$928( on rounding off to the nearest dollar).
d).Let the value of the doll double in t years from 1975. Then 2*245 = 245 e0.037t or, e0.037t = 2. Now, on taking natural log of both the sides, we get 0.037t = ln 2 = 0.69314718 so that t =0.69314718 /0.037 = 19.7 years (on rounding off to the nearest tenth).Thus, the value of the doll will double after 19.7 years.
e). Let the value of the doll become $ 1327 in t years from 1975. Then 1327 = 245 e0.037t or, e0.037t =1327/245 . Now, on taking natural log of both the sides, we get 0.037t = ln 1327 -ln 245 = 7.190676034-5.501258211 = 1.689417823 so that t =1.689417823 /0.037 = 45.7 (on rounding off to the nearest tenth). Thus, the the value of the doll will become $ 1327 after 45.7 years.
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