Question

A doll sold for $209 in 1977 and was sold again in 1989 for $467. 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 V0=209
b) Find the Exponential growth function in terms of t, where t is the number of years since 1977
c) Estimate the value of the doll in 2009.
d) What is the doubling time for the value of the doll to the nearest tenth of a year?
e) Find the amount of time after which the value of the doll with be $2221

Answer 1

Explanation:

the exponential growth function is

f(t) = v0 × (1 + k)^t

1977 to 1989 is 12 years (t).

a)

so, we have

467 = 209 × (1 + k)¹²

467/209 = (1 + k)¹²

12th root(467/209) = 1 + k = 1.069295033...

k = 0.069295033...

b)

f(t) = 209 × 1.069295033...^t

c)

2009 = 32 years after 1977

f(32) = 209 × 1.069295033...³² = $1,783.478472... ≈

≈ $1,783.48

d)

double the value of 209 is 418.

418 = 209 × 1.069295033...^t

2 = 1.069295033...^t

t = log1.069295033...(2) =

= log(2)/log(1.069295033...) = 10.34554457... ≈

≈ 10.3 years

e)

2221 = 209 × 1.069295033...^t

2221/209 = 1.069295033...^t

t = log1.069295033...(2221/209) =

= log(2221/209)/log(1.069295033...) =

= 35.27452616... years ≈

≈ 35.3 years

35 years would mean end of 2012. so, 35.3 years would be early 2013.