Step 1 :Let the height of the palm tree be denoted as \(h\).
Step 2 :Using similar triangles, we can set up the proportion: \(\frac{h}{5} = \frac{14}{4}\).
Step 3 :Solve for \(h\) by cross-multiplying: \(h = 5 \times \frac{14}{4}\).
Step 4 :Calculate the height of the palm tree: \(h = 5 \times 3.5 = 17.5\).
Step 5 :Final Answer: The height of the palm tree is \(\boxed{17.5}\) feet.