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). $V(t)=30000 \times e^{0.147 t}$
(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 .
dollar(s) per year.
(Round to the nearest million as needed.)

Step 1 ：The value of the painting is given by the function \(V(t)=30000 \times e^{0.147 t}\), 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 \(V'(t)=30000 \times 0.147 \times e^{0.147 t}\).

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 \(\boxed{\$1,589,924,965.64}\) and the rate of change of the painting's value in 2024 is approximately \(\boxed{\$233,718,969.95}\) per year.