Step 1: Understand the concept

At the same time of the day, the angle of elevation of the sun is the same for all objects. Thus, the triangles formed by the objects and their shadows are similar, meaning the ratio of height to shadow length is constant.

$\frac{\text{Height}_1}{\text{Shadow}_1} = \frac{\text{Height}_2}{\text{Shadow}_2}$

Step 2: Set up the proportion

Let the height of the pole be $H$ feet. Given that the man's height is $6$ feet, his shadow is $4$ feet, and the pole's shadow is $36$ feet.

$\frac{H}{36} = \frac{6}{4}$

Step 3: Solve for the height of the pole ($H$)

Cross-multiplying or simplifying the fraction on the right side: $\frac{6}{4} = 1.5$. Then multiply by $36$.

$H = 36 \times \frac{6}{4} = 36 \times 1.5 = 54$