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 $V$, for which $V(t)$ is the painting's value, in dollars, $t$ years after 1950 . $V(t)=26000 \times e^{0.155 t}$ (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 2026 . (Round to the nearest million as needed.)

Step 1 ：We know that the value of the painting in 1950 was $26,000 and in 2004 it was $112,207,000. We can use these two points to find the exponential growth rate. The general form of an exponential function is \(V(t) = V_0 * e^{kt}\), where \(V_0\) is the initial value, \(k\) is the growth rate, and \(t\) is time. We can plug in the values we know and solve for \(k\).

Step 2 ：Let's denote the value of the painting in 1950 as \(V_{1950} = 26000\), and the value in 2004 as \(V_{2004} = 112207000\). The corresponding years are \(t_{1950} = 1950\) and \(t_{2004} = 2004\).

Step 3 ：By substituting these values into the exponential growth function, we can solve for \(k\), which gives us \(k = 0.15500007630916599\).

Step 4 ：Once we have the growth rate, we can use it to predict the value of the painting in 2026. We just need to plug in the year 2026 into our exponential function and calculate the value. Let's denote the year 2026 as \(t_{2026} = 2026\).

Step 5 ：Substituting \(t_{2026}\) into the exponential growth function, we get \(V_{2026} = 3395977963.783134\).

Step 6 ：The exponential growth rate \(k\) is approximately \(\boxed{0.155}\). The value of the painting in 2026 will be approximately \(\boxed{\$3,395,977,963.78}\).