To find the height of the flagpole (\(x\)) and the distance between Ami and the top of the flagpole (\(y\)), we can use some basic trigonometry.
-
Calculating the height of the flagpole \(x\):
- According to the problem, Ami is looking at the flagpole from a distance of 50 feet at an angle of elevation of 35 degrees.
- We can use the tangent function which relates the angle of elevation to the opposite side (height of the flagpole \(x\)) and the adjacent side (distance from Ami to the flagpole).
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{50} \]
Plugging in the angle:
\[ \tan(35^\circ) = \frac{x}{50} \]
Now, solving for \(x\):
\[ x = 50 \cdot \tan(35^\circ) \]
Using a calculator:
\[ \tan(35^\circ) \approx 0.7002 \] \[ x \approx 50 \cdot 0.7002 \approx 35.01 \text{ feet} \]
-
Calculating the distance \(y\) from Ami to the top of the flagpole:
- To find \(y\), the distance to the top of the flagpole, we can use the sine function:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{y} \]
Rearranging gives:
\[ y = \frac{x}{\sin(35^\circ)} \]
We already found \(x \approx 35.01\) feet, so substituting:
\[ y = \frac{35.01}{\sin(35^\circ)} \]
Finding \(\sin(35^\circ) \approx 0.5736\):
\[ y \approx \frac{35.01}{0.5736} \approx 61.04 \text{ feet} \]
In conclusion, the calculations yield:
- Height of the flagpole \(x \approx 35.01\) feet
- Distance from Ami to the top of the flagpole \(y \approx 61.04\) feet
Thus, the answer corresponds to x = 35.01 feet and y = 61.04 feet.