To determine the height of the flagpole (\(x\)) and the distance between Deshaun and the top of the flagpole (\(y\)), we can use trigonometry.
Step 1: Calculate the height of the flagpole (\(x\))
Using the angle of elevation and the distance from Deshaun to the flagpole, we can set up a right triangle where:
- The adjacent side is the distance from Deshaun to the base of the flagpole, which is 50 feet.
- The opposite side is the height of the flagpole (\(x\)).
- The angle of elevation is 35 degrees.
Using the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
we can substitute into the equation:
\[ \tan(35^\circ) = \frac{x}{50} \]
Solving for \(x\):
\[ x = 50 \times \tan(35^\circ) \]
Using a calculator:
\[ \tan(35^\circ) \approx 0.7002 \]
So:
\[ x \approx 50 \times 0.7002 \approx 35.01 \text{ feet} \]
Step 2: Calculate the distance to the top of the flagpole (\(y\))
To find the distance \(y\), which is the hypotenuse of the right triangle, we can use the cosine function:
Using the cosine function:
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
We rearrange it to find the hypotenuse:
\[ \text{hypotenuse} = \frac{\text{adjacent}}{\cos(\theta)} \]
Substituting in our values:
\[ y = \frac{50}{\cos(35^\circ)} \]
Using a calculator for \(\cos(35^\circ)\):
\[ \cos(35^\circ) \approx 0.8192 \]
Thus:
\[ y \approx \frac{50}{0.8192} \approx 61.00 \text{ feet} \]
Conclusion
- The height of the flagpole (\(x\)) is approximately 35.01 feet.
- The distance from Deshaun to the top of the flagpole (\(y\)) is approximately 61.00 feet.
1 answer