To find the horizontal distance from the fan to the stadium, we can use the tangent of the angle of elevation. The relationship can be described using the following formula from trigonometry:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In this case:
- \( \theta = 7^\circ \) (the angle of elevation)
- The "opposite" side is the height of the blimp, which is 400 feet.
- The "adjacent" side is the horizontal distance from the fan to the stadium, which we will denote as \( d \).
Substituting the known values into the equation:
\[ \tan(7^\circ) = \frac{400}{d} \]
Now, to find \( d \), we can rearrange the equation:
\[ d = \frac{400}{\tan(7^\circ)} \]
Next, we calculate \( \tan(7^\circ) \).
Using a calculator:
\[ \tan(7^\circ) \approx 0.1228 \]
Now we can proceed with the calculation for \( d \):
\[ d = \frac{400}{0.1228} \approx 3254.43 \]
Rounding to the nearest foot, the horizontal distance \( d \) is approximately:
\[ d \approx 3254 \text{ feet} \]
Thus, the fan is standing about 3254 feet away from the stadium.