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). $V(t)=30000\times e^{0.147t}$
(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 ：The value of the painting is given by the function $V(t)=30000\times e^{0.147t}$, where $t$ is the number of years since 1950.

Step 2 ：To find the value of the painting in 2024, we need to substitute $t=2024-1950=74$ into the function.

Step 3 ：The value of the painting in 2024 is approximately $1,589,924,965.64.

Step 4 ：To find the rate of change of the painting's value in 2024, we need to find the derivative of the function $V(t)$ and then substitute $t=74$ into the derivative.

Step 5 ：The derivative of $V(t)$ is $V^{\prime }(t)=30000\times 0.147\times e^{0.147t}$.

Step 6 ：The rate of change of the painting's value in 2024 is approximately $233,718,969.95 per year.

Step 7 ：Final Answer: The value of the painting in 2024 is approximately $\boxed{{\displaystyle \mathrm{\$}1,589,924,965.64}}$ and the rate of change of the painting's value in 2024 is approximately $\boxed{{\displaystyle \mathrm{\$}233,718,969.95}}$ per year.