Question

Joseph is looking up at a flag. the base of the flag pole is 60 feet away from him at an angle of elevation from ground level of  29° .   What is the height of the flagpole (x) and the distance between Joseph and the top of the flagpole (y)? 

Answer 1

To determine the height of the flagpole (x) and the distance between Joseph and the top of the flagpole (y), we can use trigonometric relationships.

### Step 1: Calculate the height of the flagpole (x)

Using the tangent function, which relates the angle of elevation to the opposite (height of the pole) and adjacent (distance from Joseph to the pole) sides of the right triangle formed, we have:

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

In this case, the angle \(\theta\) is \(29^\circ\), the adjacent side is the distance from Joseph to the flagpole (60 feet), and the opposite side is the height of the flagpole (x):

\[
\tan(29^\circ) = \frac{x}{60}
\]

We can solve for \(x\):

\[
x = 60 \cdot \tan(29^\circ)
\]

Using a calculator to find \(\tan(29^\circ)\):

\[
\tan(29^\circ) \approx 0.5543
\]

Now substitute this value into the equation for \(x\):

\[
x \approx 60 \cdot 0.5543 \approx 33.26 \text{ feet}
\]

### Step 2: Calculate the distance between Joseph and the top of the flagpole (y)

To find the distance \(y\) from Joseph to the top of the flagpole, we can use the sine function in the right triangle:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

Applying it to our triangle:

\[
\sin(29^\circ) = \frac{x}{y}
\]

We can rearrange this to solve for \(y\):

\[
y = \frac{x}{\sin(29^\circ)}
\]

Using a calculator to find \(\sin(29^\circ)\):

\[
\sin(29^\circ) \approx 0.4848
\]

Substituting \(x\) into the equation for \(y\):

\[
y \approx \frac{33.26}{0.4848} \approx 68.68 \text{ feet}
\]

### Summary of Results
- The height of the flagpole \(x\) is approximately **33.26 feet**.
- The distance \(y\) from Joseph to the top of the flagpole is approximately **68.68 feet**.