You are looking up at a flag that is 25 meters away from you at a 40-degree angle of elevation with the ground. To the nearest meter, what is the height of the flagpole, x, and the distance between you and the top of the flag pole, y? Round to the hundredths place as needed.

X=____meters and Y=_____meters

1 answer

To find the height of the flagpole \( x \) and the distance from you to the top of the flagpole \( y \), we can use some basic trigonometry.

  1. Find the height of the flagpole \( x \):

Given:

  • Distance from you to the flagpole (horizontal distance, adjacent side) = 25 meters
  • Angle of elevation = 40 degrees

Using the tangent function, which is defined as:

\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

we can express the height of the flagpole \( x \) as:

\[ \tan(40^\circ) = \frac{x}{25} \]

Rearranging this equation to solve for \( x \):

\[ x = 25 \cdot \tan(40^\circ) \]

Now, calculate \( x \):

Using a calculator, we find:

\(\tan(40^\circ) \approx 0.8391\)

So:

\[ x \approx 25 \cdot 0.8391 \approx 20.9775 \]

Rounding to the nearest meter:

\[ x \approx 21 \text{ meters} \]

  1. Find the distance \( y \) from you to the top of the flagpole:

To find the distance \( y \), we can use the cosine function:

\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]

Here, the hypotenuse \( y \) relates to the height of the flagpole and the distance to it:

\[ \cos(40^\circ) = \frac{25}{y} \]

Rearranging to find \( y \):

\[ y = \frac{25}{\cos(40^\circ)} \]

Calculating \( \cos(40^\circ) \):

\(\cos(40^\circ) \approx 0.7660\)

Substituting this value:

\[ y \approx \frac{25}{0.7660} \approx 32.65 \]

Rounding to the hundredths place:

\[ y \approx 32.65 \text{ meters} \]

Final Answers:

  • \( x = 21 \) meters
  • \( y \approx 32.65 \) meters