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Sagot :
Sure, let's solve the problem step-by-step to find the angle of elevation of the sun from the ground to the top of a tree.
### Problem Recap:
We are given:
- The height of a tree: [tex]\( h = 10 \)[/tex] yards
- The length of the shadow cast by the tree: [tex]\( s = 14 \)[/tex] yards
We need to find the angle of elevation, [tex]\(\theta\)[/tex], of the sun from the ground to the top of the tree, and round it to the nearest degree.
### Step-by-Step Solution:
1. Identify the tangent relationship:
- The angle of elevation can be found using the tangent function in trigonometry:
[tex]\[ \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \][/tex]
- In this scenario:
[tex]\[ \tan(\theta) = \frac{\text{height of the tree}}{\text{length of the shadow}} = \frac{10}{14} \][/tex]
2. Calculate the angle using the arctangent function:
- The arctangent (or inverse tangent) function will give us the angle:
[tex]\[ \theta = \arctan\left(\frac{10}{14}\right) \][/tex]
3. Convert [tex]\(\theta\)[/tex] from radians to degrees:
- Trigonometric functions often give angles in radians, so we convert it to degrees using the conversion factor:
[tex]\[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \][/tex]
- Suppose our result in radians is [tex]\(\theta_{\text{radians}} = 0.6202494859828215\)[/tex] (as obtained through calculation).
4. Convert the radians to degrees:
- Using the conversion mentioned, we get:
[tex]\[ \theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi} \approx 35.53767779197438 \text{ degrees} \][/tex]
5. Round the angle to the nearest degree:
- Rounding 35.53767779197438 to the nearest whole number, we get:
[tex]\[ \theta_{\text{rounded}} = 36 \text{ degrees} \][/tex]
### Final Answer:
- The angle of elevation of the sun from the ground to the top of the tree is:
[tex]\[ \boxed{36 \text{ degrees}} \][/tex]
This means the correct option from the choices given is:
- 36 degrees
### Problem Recap:
We are given:
- The height of a tree: [tex]\( h = 10 \)[/tex] yards
- The length of the shadow cast by the tree: [tex]\( s = 14 \)[/tex] yards
We need to find the angle of elevation, [tex]\(\theta\)[/tex], of the sun from the ground to the top of the tree, and round it to the nearest degree.
### Step-by-Step Solution:
1. Identify the tangent relationship:
- The angle of elevation can be found using the tangent function in trigonometry:
[tex]\[ \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \][/tex]
- In this scenario:
[tex]\[ \tan(\theta) = \frac{\text{height of the tree}}{\text{length of the shadow}} = \frac{10}{14} \][/tex]
2. Calculate the angle using the arctangent function:
- The arctangent (or inverse tangent) function will give us the angle:
[tex]\[ \theta = \arctan\left(\frac{10}{14}\right) \][/tex]
3. Convert [tex]\(\theta\)[/tex] from radians to degrees:
- Trigonometric functions often give angles in radians, so we convert it to degrees using the conversion factor:
[tex]\[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \][/tex]
- Suppose our result in radians is [tex]\(\theta_{\text{radians}} = 0.6202494859828215\)[/tex] (as obtained through calculation).
4. Convert the radians to degrees:
- Using the conversion mentioned, we get:
[tex]\[ \theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi} \approx 35.53767779197438 \text{ degrees} \][/tex]
5. Round the angle to the nearest degree:
- Rounding 35.53767779197438 to the nearest whole number, we get:
[tex]\[ \theta_{\text{rounded}} = 36 \text{ degrees} \][/tex]
### Final Answer:
- The angle of elevation of the sun from the ground to the top of the tree is:
[tex]\[ \boxed{36 \text{ degrees}} \][/tex]
This means the correct option from the choices given is:
- 36 degrees
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