To solve the problem, we can use trigonometry, specifically the tangent function. The situation can be modeled as a right triangle where:
- The height of the blimp (opposite side) = 400 feet
- The distance from the fan to the stadium (adjacent side) = \(d\) feet
- The angle of elevation = 7 degrees
The relationship between the angle, the opposite side, and the adjacent side is given by the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In our case, we have:
\[ \tan(7^\circ) = \frac{400}{d} \]
We can rearrange this equation to solve for \(d\):
\[ d = \frac{400}{\tan(7^\circ)} \]
Now, we can calculate \(\tan(7^\circ)\). Using a calculator:
\[ \tan(7^\circ) \approx 0.122784 \]
Now substituting this back into the equation for \(d\):
\[ d \approx \frac{400}{0.122784} \]
Calculating the value:
\[ d \approx 3255.72 \]
Rounding this to the nearest foot, we find:
\[ d \approx 3256 \text{ feet} \]
Thus, the fan is standing approximately 3256 feet away from the stadium.