To determine the height of the tree, we can use trigonometric relationships, specifically the tangent function. Here’s a detailed, step-by-step solution:
1. Understand the problem: We are given the angle of elevation (68°) and the length of the shadow (14.3 meters). We need to find the height of the tree.
2. Using the tangent function:
- The tangent of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side (which in this case is the height of the tree) to the length of the adjacent side (which is the length of the shadow).
- Therefore, [tex]\( \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \)[/tex].
3. Set up the equation:
- Here, [tex]\(\text{angle} = 68°\)[/tex], [tex]\(\text{adjacent} = 14.3 \, \text{meters}\)[/tex], and the [tex]\(\text{opposite}\)[/tex] is the height of the tree, which we will denote as [tex]\( h \)[/tex].
- So, [tex]\( \tan(68°) = \frac{h}{14.3} \)[/tex].
4. Solve for [tex]\( h \)[/tex]:
- Rearranging the equation, we get:
[tex]\[
h = \tan(68°) \times 14.3
\][/tex]
5. Calculate the height:
- First, convert the angle from degrees to radians, since trigonometric functions typically use radians. (Note: degrees to radians conversion is not shown here, but it’s used internally in calculations.)
- Use the tangent value for 68°.
- Calculate the height:
[tex]\[
h \approx \tan(68°) \times 14.3 \approx 35.39374200385304 \, \text{meters}
\][/tex]
6. Round the result to the nearest tenth:
- The height [tex]\( h \approx 35.4 \, \text{meters} \)[/tex] when rounded to the nearest tenth.
So, the height of the tree, to the nearest tenth of a meter, is:
[tex]\[ 35.4 \, \text{meters} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{35.4 \, \text{meters}} \][/tex]
