Question

A building is 60 feet high, and from a distance the angle of elevation to the top of the building is 40 degrees. From a closer point the angle of elevation is 70 degrees. Find the distance between the two points on the ground?

Answer 1

To solve this problem, we can use trigonometry, specifically the tangent function, which relates angles of a right triangle to the ratio of the opposite side to the adjacent side.

Let \( d_1 \) be the distance from the first point to the base of the building, and let \( d_2 \) be the distance from the second point (closer) to the base of the building. The total distance between the two points on the ground is \( d_1 - d_2 \).

### Step 1: Set up the equations

For the first position (40 degrees):
\[
\tan(40^\circ) = \frac{60}{d_1}
\]
This can be rearranged to:
\[
d_1 = \frac{60}{\tan(40^\circ)}
\]

For the second position (70 degrees):
\[
\tan(70^\circ) = \frac{60}{d_2}
\]
This can be rearranged to:
\[
d_2 = \frac{60}{\tan(70^\circ)}
\]

### Step 2: Calculate \( d_1 \) and \( d_2 \)

Using approximate values for the tangent:
- \( \tan(40^\circ) \approx 0.8391 \)
- \( \tan(70^\circ) \approx 2.7475 \)

Now, calculate \( d_1 \):
\[
d_1 = \frac{60}{0.8391} \approx 71.5 \text{ feet}
\]

Next, calculate \( d_2 \):
\[
d_2 = \frac{60}{2.7475} \approx 21.8 \text{ feet}
\]

### Step 3: Find the distance between the two points

Finally, the distance between the two points on the ground is:
\[
d_1 - d_2 \approx 71.5 - 21.8 \approx 49.7 \text{ feet}
\]

Thus, the distance between the two points on the ground is approximately
\[
\boxed{49.7} \text{ feet}.
\]