Answer :
To solve the problem of finding the distance Sharon traveled between her two viewings of the eagle, we'll break the problem into simpler steps:
1. Understand the Angles:
- Initially, Sharon measures an angle of elevation of [tex]\(37^{\circ} 27^{\prime}\)[/tex].
- After moving closer, she measures an angle of [tex]\(51^{\circ} 42^{\prime}\)[/tex].
2. Convert Angles to Decimal Degrees:
- For the first angle: Convert the minutes to a fraction of a degree. [tex]\(27\)[/tex] minutes is equivalent to [tex]\(\frac{27}{60}\)[/tex] degrees. So, the first angle in decimal is:
[tex]\[
37 + \frac{27}{60} = 37.45\, \text{degrees}
\][/tex]
- For the second angle: Convert the minutes to a fraction of a degree. [tex]\(42\)[/tex] minutes is equivalent to [tex]\(\frac{42}{60}\)[/tex] degrees. So, the second angle in decimal is:
[tex]\[
51 + \frac{42}{60} = 51.7\, \text{degrees}
\][/tex]
3. Calculate the Change in Angle of Elevation:
- Subtract the initial angle from the new angle to find out how much the angle of elevation increased.
[tex]\[
51.7 - 37.45 = 14.25\, \text{degrees}
\][/tex]
The change in the angle of elevation, after Sharon moved closer to the tree, is [tex]\(14.25\)[/tex] degrees. This change in angle is useful to determine the distance Sharon moved, but the details of the distance calculation are not included here as we don't have information on distances or tree height.
To fully solve for distance, information about the height of the tree or an initial distance would be required. However, the calculation of angle change is clear and accurate as [tex]\(14.25\)[/tex] degrees.
1. Understand the Angles:
- Initially, Sharon measures an angle of elevation of [tex]\(37^{\circ} 27^{\prime}\)[/tex].
- After moving closer, she measures an angle of [tex]\(51^{\circ} 42^{\prime}\)[/tex].
2. Convert Angles to Decimal Degrees:
- For the first angle: Convert the minutes to a fraction of a degree. [tex]\(27\)[/tex] minutes is equivalent to [tex]\(\frac{27}{60}\)[/tex] degrees. So, the first angle in decimal is:
[tex]\[
37 + \frac{27}{60} = 37.45\, \text{degrees}
\][/tex]
- For the second angle: Convert the minutes to a fraction of a degree. [tex]\(42\)[/tex] minutes is equivalent to [tex]\(\frac{42}{60}\)[/tex] degrees. So, the second angle in decimal is:
[tex]\[
51 + \frac{42}{60} = 51.7\, \text{degrees}
\][/tex]
3. Calculate the Change in Angle of Elevation:
- Subtract the initial angle from the new angle to find out how much the angle of elevation increased.
[tex]\[
51.7 - 37.45 = 14.25\, \text{degrees}
\][/tex]
The change in the angle of elevation, after Sharon moved closer to the tree, is [tex]\(14.25\)[/tex] degrees. This change in angle is useful to determine the distance Sharon moved, but the details of the distance calculation are not included here as we don't have information on distances or tree height.
To fully solve for distance, information about the height of the tree or an initial distance would be required. However, the calculation of angle change is clear and accurate as [tex]\(14.25\)[/tex] degrees.
