The values of certain types of collectibles can fluctuate greatly over time. Sup
ID: 3204190 • Letter: T
Question
The values of certain types of collectibles can fluctuate greatly over time. Suppose that the value of a particular limited edition crystal rabbit is found to be able to be modeled by the function v (t) = - 0.01t^4 + 0.41 t^2 - 6.28 t^2 + 41.65t + 70 for 0 lessthanorequalto t lessthanorequalto 20 where v (t) dollars t is the number of years after the rabbit was released, and t = 0 corresponds to the year 1996 What was the value of the rabbit in the year 2001? What was the value of the rabbit in the year 2016? What was theinstantaneous rate of change ofthe value oftherabbit in the year 1999? What was the instantaneous rate of change of the value oftherabbit in the year 2016? Use your answers from parts a-d to ESTIMATE the value ofthe rabbit in 2017Explanation / Answer
Solution:-
V(t) = - 0.01t4 + 0.41t3 - 6.28t2 + 41.65t + 70
a) The value of the rabit in the year 2001 is 167
t = 2001 - 1996 = 5
V(t) = - 0.01(5)4 + 0.41(5)3 - 6.28(5)2 + 41.65 × 5 + 70
V(t) = - 6.25 + 51.25 - 157 + 208.25 + 70
V(t) = 166.25
b) The value of the rabit in the year 2016 is 71.
t = 2016 - 1996 = 20
V(t) = - 0.01(20)4 + 0.41(20)3 - 6.28(20)2 + 41.65 × 20 + 70
V(t) = - 1,600 + 3,280 - 2512 + 833 + 70
V(t) = 71
c) The instantaneous rate of change of the value of the rabbit in the year 1999 is 14.
Rate of change = - 0.04t3 + 1.23t2 - 12.56t + 41.65
t = 1999 - 1996 = 3
Rate of change = - 0.04t3 + 1.23t2 - 12.56t + 41.65
Rate of change = - 0.04(3)3 + 1.23(3)2 - 12.56 × 3 + 41.65
Rate of change = - 1.08 + 11.07 - 37.68 + 41.65
Rate of change = 13.99
Rate of change = 14
d)
The instantaneous rate of change of the value of the rabbit in the year 2016 is - 38.
Rate of change = - 0.04t3 + 1.23t2 - 12.56t + 41.65
t = 2016 - 1996 = 20
Rate of change = - 0.04t3 + 1.23t2 - 12.56t + 41.65
Rate of change = - 0.04(20)3 + 1.23(20)2 - 12.56 × 20 + 41.65
Rate of change = - 320 + 492 - 251.2 + 41.65
Rate of change = - 37.55
Rate of change = - 38
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