Sure, let's solve this step by step using the given angles and distances.
### Step 1: Identify Known Values
- Distance between the students: [tex]\( \overline{AB} = 1 \text{ yard} \)[/tex]
- Angle measured by Student A ([tex]\(\angle A\)[/tex]): [tex]\( 57^\circ \)[/tex]
- Angle measured by Student B ([tex]\(\angle B\)[/tex]): [tex]\( 46^\circ \)[/tex]
### Step 2: Calculate the Third Angle ([tex]\(\angle C\)[/tex])
Since the angles in a triangle sum to [tex]\( 180^\circ \)[/tex]:
[tex]\[
\angle C = 180^\circ - \angle A - \angle B = 180^\circ - 57^\circ - 46^\circ = 77^\circ
\][/tex]
### Step 3: Using the Law of Sines to Find [tex]\( \overline{AT} \)[/tex]
The law of sines states:
[tex]\[
\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}
\][/tex]
Here, [tex]\(a = \overline{AT}\)[/tex], [tex]\(b = \overline{BT}\)[/tex], and [tex]\(c = \overline{AB} = 1 \text{ yard}\)[/tex].
We need to find [tex]\( \overline{AT} \)[/tex]:
[tex]\[
\overline{AT} = \frac{\sin(A) \cdot \overline{AB}}{\sin(C)} = \frac{\sin(57^\circ) \cdot 1}{\sin(77^\circ)}
\][/tex]
The given numerical answer reveals:
[tex]\[ \overline{AT} \approx 0.8607 \text{ yards} \][/tex]
### Step 4: Calculate the Height of the Tree ([tex]\( h \)[/tex])
The height [tex]\( h \)[/tex] of the tree can be found using Student A's angle and the distance [tex]\( \overline{AT} \)[/tex]:
[tex]\[
h = \overline{AT} \cdot \sin(57^\circ)
\][/tex]
The given numerical answer reveals:
[tex]\[ h \approx 0.7219 \text{ yards} \][/tex]
### Conclusion
The height of the tree is approximately [tex]\( 0.7219 \text{ yards} \)[/tex]. However, according to the provided multiple-choice options (which are significantly larger values than our calculated height), none of these provided options are close to the value we have derived.
Therefore, let's state that if the question had these multiple choice options, then the correct method would guide you to a more accurate realistic context calculator because the height calculated does not match any options: 3.0 yards, 3.2 yards, 3.8 yards, or 4.4 yards.
