In 2004 , an art collector paid $\mathrm{\$}84,053,000$ for a particular painting. The same painting sold for $\mathrm{\$}30,000$ 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 . $V(t)=30000\times e^{0.147t}$
(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 .
$\mathrm{\$}1,590,000,000$
(Round to the nearest million as needed.)
c) Estimate the rate of change of the painting's value in 2024.
dollar(s) per year.
(Round to the nearest million as needed.)

Step 1 ：Let's denote the value of the painting in 1950 as $V_0$, which is $30,000. The value of the painting in 2004 is $84,053,000, which is 54 years after 1950. We can denote this as $V(54)$.

Step 2 ：We know that the value of the painting follows an exponential growth function, which can be written as $V(t)=V_0\cdot e^{kt}$, where $k$ is the exponential growth rate, $t$ is the time in years after 1950, and $e$ is the base of the natural logarithm.

Step 3 ：We can substitute $V_0$ and $V(54)$ into the equation to solve for $k$: $84,053,000=30,000\cdot e^{54k}$.

Step 4 ：Divide both sides by 30,000 to isolate the exponential term: $e^{54k}=84,053,000/30,000$.

Step 5 ：Take the natural logarithm of both sides to solve for $k$: $54k=\ln (84,053,000/30,000)$.

Step 6 ：Solve for $k$ by dividing both sides by 54: $k=\ln (84,053,000/30,000)/54$.

Step 7 ：Calculate the value of $k$ to three decimal places: $k=\boxed{{\displaystyle 0.147}}$.

Step 8 ：Substitute $k$ into the exponential growth function to get the function $V(t)$: $V(t)=30,000\cdot e^{0.147t}$.

Step 9 ：To predict the value of the painting in 2024, which is 74 years after 1950, substitute $t=74$ into the function $V(t)$: $V(74)=30,000\cdot e^{0.147\cdot 74}$.

Step 10 ：Calculate the value of $V(74)$ and round to the nearest million: $V(74)=\boxed{{\displaystyle \mathrm{\$}1,590,000,000}}$.

Step 11 ：To estimate the rate of change of the painting's value in 2024, we need to find the derivative of the function $V(t)$: $V^{\prime }(t)=30,000\cdot 0.147\cdot e^{0.147t}$.

Step 12 ：Substitute $t=74$ into the derivative function $V^{\prime }(t)$ to get the rate of change in 2024: $V^{\prime }(74)=30,000\cdot 0.147\cdot e^{0.147\cdot 74}$.

Step 13 ：Calculate the value of $V^{\prime }(74)$ and round to the nearest million: $V^{\prime }(74)=\boxed{{\displaystyle \mathrm{\$}233,000,000}}$ per year.