The values of certain types of collectibles can often fluctuate greatly over tim

Question

The values of certain types of collectibles can often fluctuate greatly over time. Suppose the by value of a particular limited edition ceramic Donald Trump garden gnome is found to be able to be modeled is the function V(t) = -0.01t^4 + 0.42t^3 + 40.78t + 59 for 0 lessthanorequalto t lessthanorequalto 20 where V(t) is in dollars, I is the number of years after the gnome was released, and t = 0 corresponds to the year 1996. a) What was the value of the gnome in the year 2001? b) What was the value of the gnome in the year 2016? c) What was the instantaneous rate of change of the value of the gnome in the year 1999? d) What was the instantaneous rate of change of the value of the gnome in the year 2016? e) Use your answers from parts a-d to ESTIMATE the value of the gnome in 2017.

Explanation / Answer

v(t)=-0.01*t^4+0.42*t^3-6.43*t^2+40.78*t+59

a)

at year  2001 means t= 5

so substitute the value of t=5 in above equation we can get

v(5) =   148.4000 $

b) year 2016 means t= 20

substituting value of t =20 in v(t) we get

v(20)=  62.6000

c)

instantaneous change can be calculated by derivative

v' (t) =- t^3/25 + (63*t^2)/50 - (643*t)/50 + 2039/50

now year 1999 means t= 2

thus v'(2)= 19.7800

d)

year 2016 means t= 20

v'(20)= -32.4200

e)

now we have to esimate v(21) ie for year 2017

v(21)= v(20)+ instantaneous change in year 2016 = 60.6-32.42=28.18

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