A radio tower is 200 feet tall. How far is an observer from the base of the tower if the angle of their line of sight from the ground to the top of the tower is 65°

Question

? Round to the nearest
hundredth.



(1 point)
The observer is
feet away.

Answer 1

To find the distance from the observer to the base of the tower, we can use the tangent of the angle. In this case, we have:

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

Here, the height of the tower (200 feet) is the opposite side, and the distance from the observer to the base of the tower is the adjacent side. Let's denote the distance from the observer to the base of the tower as \(d\).

Given:

• Height of the tower (opposite) = 200 feet
• Angle (\(\theta\)) = 65°

We can rearrange the tangent formula to solve for \(d\):

\[ d = \frac{\text{opposite}}{\tan(\theta)} = \frac{200}{\tan(65^\circ)} \]

Now, we calculate \(\tan(65^\circ)\):

Using a calculator, we find:

\[ \tan(65^\circ) \approx 2.1445 \]

Now substitute this back into the equation for \(d\):

\[ d = \frac{200}{2.1445} \approx 93.24 \]

Therefore, the observer is approximately 93.24 feet away from the base of the tower.

Final answer: The observer is 93.24 feet away.

Answer 2

no

Answer 3

Let's go through the calculation step by step to ensure we find the correct distance.

1. We know the height of the tower (the opposite side of the triangle) is 200 feet.
2. The angle of elevation to the top of the tower is 65°.

Using the tangent function:

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

In this case:

\[ \tan(65^\circ) = \frac{200}{d} \]

We want to find \(d\), the distance from the observer to the base of the tower. Rearranging the formula gives us:

\[ d = \frac{200}{\tan(65^\circ)} \]

Now, we need to compute \(\tan(65^\circ)\):

Using a calculator:

\[ \tan(65^\circ) \approx 2.1445 \]

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

\[ d = \frac{200}{2.1445} \approx 93.24 \]

This result aligns with my previous answer. However, let's confirm it with a more precise calculation.

Using a calculator directly for distance computation:

1. Calculate \(\tan(65^\circ)\):

   • \(\tan(65^\circ) \approx 2.1445\) (depending on your calculator's setting, you might get a slightly different result, but it should be around this)

2. Now using that in the formula: \[ d \approx \frac{200}{2.1445} \approx 93.24 \text{ feet} \]

If this doesn't align with your expected results, please clarify any specific parameters or conditions I should consider. Otherwise, using conventional methods, the observer is correctly approximately 93.24 feet away from the base of the tower.