Question

In​ 2004, an art collector paid ​$ for a particular painting. The same painting sold for ​$ in 1950. Complete parts​ (a) through​ (d). ​a) Find the 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.

Answer 1

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:

$y=a\cdot e^{kt}$

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:

$y=a\cdot e^{kt}$

$84,275,000=27,000\times e^{54k}\\\\e^{54k}=\frac{84275000}{27000}\\\\54k=\ln (\frac{84275}{27})\\\\k=\frac{1}{54}\times \ln (\frac{84275}{27})\\\\k=0.149$

Thus, the value of k is 0.149.