In 2004, an art collector paid $\$ 112,207,000$ for a particular painting. The same painting sold for $\$ 26,000$ in 1950 . Complete parts (a) through (d). a) Find the exponential growth rate $k$, to three decimal places, and determine the exponential growth function $\mathrm{V}$, for which $\mathrm{V}(\mathrm{t})$ is the painting's value, in dollars, $t$ years after 1950 . \[ V(t)= \] (Type an expression. Type integers or decimals for any numbers in the expression. Round to three decimal places as needed.)

Step 1 ：The exponential growth function is given by the formula: \(V(t) = P * e^{kt}\), where \(V(t)\) is the value of the painting t years after 1950, P is the initial value of the painting in 1950, k is the exponential growth rate, and t is the number of years after 1950.

Step 2 ：We know that: P = $26,000 and \(V(t)\) = $112,207,000 in 2004, which is 54 years after 1950.

Step 3 ：We can substitute these values into the formula and solve for k: 112207000 = 26000 * e^(54k).

Step 4 ：Solving this equation for k, we get k = 0.15500007630916599.

Step 5 ：The question asks for the value of k to three decimal places. Therefore, we need to round the calculated value of k to three decimal places. The rounded value of k is 0.155.

Step 6 ：Final Answer: The exponential growth rate \(k\) is \(\boxed{0.155}\). The exponential growth function \(V(t)\) is \(V(t) = 26000 * e^{0.155t}\).