Answer:
The value of k is 0.149.
Step-by-step explanation:
The complete question is:
In 2004, an art collector paid $84,275,000 for a particular painting. The same painting sold for $27,000 in 1950. Complete part (a) through (d)? (a) Find exponential growth rate k, to three decimal places, and determine the exponential growth function V, for which V(t) is the painting’s value, in dollars, t years after 1950.
Solution:
The exponential growth function can be expressed as follows:
[tex]y=a\cdot e^{kt}[/tex]
y = Final value after t years
a = initial value
It is provided that:
y = $84,275,000
a = $27,000
The value of t from 1950 to 2004 is, t = 54
Compute the value of k as follows:
[tex]y=a\cdot e^{kt}[/tex]
[tex]84,275,000=27,000\times e^{54k}\\\\e^{54k}=\frac{84275000}{27000}\\\\54k=\ln (\frac{84275}{27})\\\\k=\frac{1}{54}\times \ln (\frac{84275}{27})\\\\k=0.149[/tex]
Thus, the value of k is 0.149.
