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.)
Solution
Step 1 ：Given that the initial value of the painting in 1950, denoted as \(P_0\), is $30,000, the value of the painting in 2004, denoted as \(V_t\), is $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 \(V(t) = P_0 * e^{kt}\), where \(k\) is the growth rate, to solve for \(k\).
Step 3 ：Substituting the given values into the equation, we get \(84053000 = 30000 * e^{54k}\).
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 \(V(t) = 30000 * e^{0.147t}\).
Step 6 ：\(\boxed{The exponential growth rate k is approximately 0.147 and the exponential growth function V(t) is V(t) = 30000 * e^{0.147t}}\)
