Alright, let's walk through the solution step-by-step to find the height of the tree, [tex]\( h \)[/tex].
1. Determine the given angles:
- Angle [tex]\(A\)[/tex]: [tex]\(57^\circ\)[/tex]
- Angle [tex]\(B\)[/tex]: [tex]\(46^\circ\)[/tex]
2. Calculate the third angle [tex]\(C\)[/tex] in the triangle formed by the students and the tree:
[tex]\[
C = 180^\circ - A - B = 180^\circ - 57^\circ - 46^\circ = 77^\circ
\][/tex]
3. Apply the Law of Sines to find the length [tex]\(AT\)[/tex]:
The law of sines states:
[tex]\[
\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}
\][/tex]
Here, [tex]\(a = 1\)[/tex] yard (the distance between the students),
[tex]\[
\frac{\sin(57^\circ)}{AT} = \frac{\sin(46^\circ)}{1 \text{ yard}} = \frac{\sin(77^\circ)}{c}
\][/tex]
To find [tex]\(AT\)[/tex], we rearrange the formula to solve for [tex]\(AT\)[/tex]:
[tex]\[
AT = \frac{\sin(A)}{\sin(C)} \times 1 \text{ yard}
\][/tex]
4. Calculate the sine values and then [tex]\(AT\)[/tex]:
[tex]\[
\sin(57^\circ) \approx 0.8387
\][/tex]
[tex]\[
\sin(77^\circ) \approx 0.9744
\][/tex]
[tex]\[
AT = \frac{\sin(57^\circ)}{\sin(77^\circ)} \approx \frac{0.8387}{0.9744} \approx 0.8607 \text{ yards}
\][/tex]
5. Determine the height of the tree [tex]\(h\)[/tex] using [tex]\(AT\)[/tex] and angle [tex]\(A\)[/tex]:
In triangle [tex]\( ATO \)[/tex] (where [tex]\(O\)[/tex] is the top of the tree),
[tex]\[
h = AT \times \sin(A)
\][/tex]
[tex]\[
h = 0.8607 \text{ yards} \times \sin(57^\circ) \approx 0.8607 \times 0.8387 \approx 0.7219 \text{ yards}
\][/tex]
So, based on the given calculations, the height of the tree [tex]\( h \)[/tex] is approximately [tex]\(0.7219\)[/tex] yards. Note that the listed choices (3.0 yards, 3.2 yards, 3.8 yards, 4.4 yards) seem significantly higher than our calculation. Therefore, it appears that there may be an error, or additional context is needed to match the expected values. However, based on the provided steps, [tex]\( h \approx 0.7219\)[/tex] yards is the height of the tree.
