Question
Question
Ella is standing on the ground, looking up toward the top of a tower. Her eye level is 5.5
feet above the ground, and her sightline forms a 21°
degree angle while looking at the top of the tower. The direct distance from her eyes to the top of the tower is 46 feet.
Which of the following correctly uses the sine ratio to find the height of the tower?
Ella is standing on the ground, looking up toward the top of a tower. Her eye level is 5.5
feet above the ground, and her sightline forms a 21°
degree angle while looking at the top of the tower. The direct distance from her eyes to the top of the tower is 46 feet.
Which of the following correctly uses the sine ratio to find the height of the tower?
Answers
GPT-4o mini
To find the height of the tower using the sine ratio, we need to establish a relationship between Ella's eye level, the height of the tower, and the angle of elevation (21 degrees).
1. Let \( h \) be the height of the tower above Ella's eye level. Since Ella's eye level is 5.5 feet above the ground, the total height \( H \) of the tower can be expressed as \( H = h + 5.5 \).
2. The sine of the angle is defined as the ratio of the opposite side to the hypotenuse in a right triangle. In this case:
- The angle of elevation \( \theta = 21^\circ \)
- The opposite side is the height \( h \) (the height from Ella's eye level to the top of the tower).
- The hypotenuse is the direct distance from Ella's eyes to the top of the tower, which is given as 46 feet.
Using the sine ratio:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{46}
\]
So, plugging the angle into the equation, we have:
\[
\sin(21^\circ) = \frac{h}{46}
\]
Therefore, to solve for \( h \), you would rearrange the equation:
\[
h = 46 \cdot \sin(21^\circ)
\]
Thus, the equation that correctly uses the sine ratio to find the height \( h \) of the tower above Ella's eye level is:
\[
h = 46 \cdot \sin(21^\circ)
\]
To find the total height of the tower above ground level, you would then add Ella's eye level:
\[
H = 46 \cdot \sin(21^\circ) + 5.5
\]
1. Let \( h \) be the height of the tower above Ella's eye level. Since Ella's eye level is 5.5 feet above the ground, the total height \( H \) of the tower can be expressed as \( H = h + 5.5 \).
2. The sine of the angle is defined as the ratio of the opposite side to the hypotenuse in a right triangle. In this case:
- The angle of elevation \( \theta = 21^\circ \)
- The opposite side is the height \( h \) (the height from Ella's eye level to the top of the tower).
- The hypotenuse is the direct distance from Ella's eyes to the top of the tower, which is given as 46 feet.
Using the sine ratio:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{46}
\]
So, plugging the angle into the equation, we have:
\[
\sin(21^\circ) = \frac{h}{46}
\]
Therefore, to solve for \( h \), you would rearrange the equation:
\[
h = 46 \cdot \sin(21^\circ)
\]
Thus, the equation that correctly uses the sine ratio to find the height \( h \) of the tower above Ella's eye level is:
\[
h = 46 \cdot \sin(21^\circ)
\]
To find the total height of the tower above ground level, you would then add Ella's eye level:
\[
H = 46 \cdot \sin(21^\circ) + 5.5
\]