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.
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: \(k = \boxed{0.147}\).

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: \(V(74) = \boxed{\$1,590,000,000}\).

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

Step 12 ：Substitute \(t = 74\) into the derivative function \(V'(t)\) to get the rate of change in 2024: \(V'(74) = 30,000 \cdot 0.147 \cdot e^{0.147 \cdot 74}\).

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