In 2004 , an art collector paid $$ 84, 053, 000$ for a particular painting. The same painting sold for $$ 30, 000$ 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. $V (t) = 30000 \times e^{0.147 t}$
(Type an expression. Type integers or decimals for any numbers in the expression. Round to three decimal places as needed.)
b) Predict the value of the painting in 2024
$$ 1, 590, 000, 000$
(Round to the nearest million as needed.)
c) Estimate the rate of change of the painting's value in 2024 .
$234, 000, 000$ dollar(s) per year.
(Round to the nearest million as needed.)
d) How long after 1950 will the value of the painting be $$ 4$ billion?
year(s)
(Lo not round until the final answer. Then round to the nearest year as needed.)

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$T h e e x p o n e n t i a l g r o w t h r a t e k i s a p p r o x i m a t e l y 0.147 a n d t h e e x p o n e n t i a l g r o w t h f u n c t i o n V (t) i s V (t) = 30000 * e^{0.147 t}$

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Step 1 ：Given that the initial value of the painting in 1950, denoted as $P_{0}$ , is $30, 000, t h e v a l u e o f t h e p a i n t i n g i n 2004, d e n o t e d a s $V_{t} $, i s$ 84,053,000, and the time $t$ from 1950 to 2004 is 54 years.

Step 2 ：We can use these values in the exponential growth function $V (t) = P_{0} * e^{k t}$ , where $k$ is the growth rate, to solve for $k$ .

Step 3 ：Substituting the given values into the equation, we get $84053000 = 30000 * e^{54 k}$ .

Step 4 ：Solving this equation for $k$ , we find that $k$ is approximately 0.147.

Step 5 ：Substituting $k = 0.147$ into the exponential growth function, we get $V (t) = 30000 * e^{0.147 t}$ .

Step 6 ： $T h e e x p o n e n t i a l g r o w t h r a t e k i s a p p r o x i m a t e l y 0.147 a n d t h e e x p o n e n t i a l g r o w t h f u n c t i o n V (t) i s V (t) = 30000 * e^{0.147 t}$

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