5.2 Section Exercise A lamppost cast a shadow of $12 \mathrm{ft}$ when the angle of elevation of the Sun is $30.2^{\circ}$. How high is the lamppost? Round to the nearest foot. The height of the lamppost is approximately $\mathrm{ft}$

Step 1 ：The problem involves trigonometry. Specifically, it involves the tangent of the angle of elevation, which is the ratio of the opposite side (the height of the lamppost) to the adjacent side (the length of the shadow). We can use the formula: \(\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}\)

Step 2 ：We know the angle and the length of the shadow, so we can solve for the height of the lamppost. The shadow length is 12 feet and the angle is 30.2 degrees.

Step 3 ：First, we need to convert the angle from degrees to radians. We use the formula \(\text{angle in radians} = \text{angle in degrees} \times \frac{\pi}{180}\). Substituting the given angle, we get \(\text{angle in radians} = 30.2 \times \frac{\pi}{180} = 0.5270894341022875\) radians.

Step 4 ：Next, we substitute the known values into the tangent formula: \(\tan(0.5270894341022875) = \frac{\text{height}}{12}\). Solving for height, we get \(\text{height} = 12 \times \tan(0.5270894341022875) = 6.984166778804404\) feet.

Step 5 ：Finally, we round the height to the nearest foot to get \(\text{height} = 7\) feet.

Step 6 ：Final Answer: The height of the lamppost is approximately \(\boxed{7}\) feet.