Right triangle trigonometry word problems for solving angles in real-world scenarios.

Problem 1: Inclined Ramp

Question: An inclined ramp has a length of 13 feet and a vertical rise of 6 feet. What angle does the bottom of the ramp make with the ground?

#### Solution:
1. Identify the given information:
- Length of the ramp (hypotenuse, \( c \)) = 13 feet
- Vertical rise (opposite side, \( o \)) = 6 feet
- We need to find the angle (\( \theta \)) that the ramp makes with the ground.

2. Visualize the problem:
- The ramp forms a right triangle where:
- The hypotenuse is the length of the ramp.
- The opposite side is the vertical rise.
- The adjacent side is the horizontal distance along the ground.

3. Use trigonometry:
- The sine function relates the opposite side to the hypotenuse:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{o}{c} = \frac{6}{13}
\]

4. Solve for \( \theta \):
- Use the inverse sine function (\( \sin^{-1} \)):
\[
\theta = \sin^{-1}\left(\frac{6}{13}\right)
\]

5. Calculate the value:
- Using a calculator:
\[
\theta \approx \sin^{-1}(0.4615) \approx 27.47^\circ
\]

#### Final Answer:
\[
\boxed{27.47^\circ}
\]

---

Problem 2: Ladder Leaning Against a Wall

Question: A 12-foot ladder leaning against a vertical wall makes an angle of 80° with the ground. How far from the base of the wall is the foot of the ladder?

#### Solution:
1. Identify the given information:
- Length of the ladder (hypotenuse, \( c \)) = 12 feet
- Angle with the ground (\( \theta \)) = 80°
- We need to find the horizontal distance (adjacent side, \( a \)) from the base of the wall to the foot of the ladder.

2. Visualize the problem:
- The ladder forms a right triangle where:
- The hypotenuse is the length of the ladder.
- The adjacent side is the horizontal distance from the wall.
- The opposite side is the height up the wall.

3. Use trigonometry:
- The cosine function relates the adjacent side to the hypotenuse:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{c}
\]
- Rearrange to solve for \( a \):
\[
a = c \cdot \cos(\theta)
\]

4. Substitute the values:
\[
a = 12 \cdot \cos(80^\circ)
\]

5. Calculate the value:
- Using a calculator:
\[
\cos(80^\circ) \approx 0.1736
\]
\[
a = 12 \cdot 0.1736 \approx 2.0832 \text{ feet}
\]

#### Final Answer:
\[
\boxed{2.08 \text{ feet}}
\]

---

Problem 3: Rectangle Diagonal Angle

Question: A rectangle is 40 inches wide and 50 inches long. What angle does the diagonal make with the width (the shorter side)?

#### Solution:
1. Identify the given information:
- Width of the rectangle (\( w \)) = 40 inches
- Length of the rectangle (\( l \)) = 50 inches
- We need to find the angle (\( \theta \)) that the diagonal makes with the width.

2. Visualize the problem:
- The diagonal of the rectangle forms a right triangle where:
- The width is one leg.
- The length is the other leg.
- The diagonal is the hypotenuse.

3. Use trigonometry:
- The tangent function relates the opposite side to the adjacent side:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{l}{w}
\]
- Substitute the values:
\[
\tan(\theta) = \frac{50}{40} = 1.25
\]

4. Solve for \( \theta \):
- Use the inverse tangent function (\( \tan^{-1} \)):
\[
\theta = \tan^{-1}(1.25)
\]

5. Calculate the value:
- Using a calculator:
\[
\theta \approx \tan^{-1}(1.25) \approx 51.34^\circ
\]

#### Final Answer:
\[
\boxed{51.34^\circ}
\]

---

Problem 4: Kite String Angle of Elevation

Question: A kite is 100 feet high in the sky when 600 feet of string is out. What angle of elevation does the kite string make with the ground?

#### Solution:
1. Identify the given information:
- Height of the kite (\( h \)) = 100 feet
- Length of the string (hypotenuse, \( c \)) = 600 feet
- We need to find the angle of elevation (\( \theta \)).

2. Visualize the problem:
- The kite string forms a right triangle where:
- The height is the opposite side.
- The hypotenuse is the length of the string.
- The adjacent side is the horizontal distance from the person to the point directly below the kite.

3. Use trigonometry:
- The sine function relates the opposite side to the hypotenuse:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{c}
\]
- Substitute the values:
\[
\sin(\theta) = \frac{100}{600} = \frac{1}{6}
\]

4. Solve for \( \theta \):
- Use the inverse sine function (\( \sin^{-1} \)):
\[
\theta = \sin^{-1}\left(\frac{1}{6}\right)
\]

5. Calculate the value:
- Using a calculator:
\[
\theta \approx \sin^{-1}(0.1667) \approx 9.59^\circ
\]

#### Final Answer:
\[
\boxed{9.59^\circ}
\]

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Summary of Answers:

1. \( \boxed{27.47^\circ} \)
2. \( \boxed{2.08 \text{ feet}} \)
3. \( \boxed{51.34^\circ} \)
4. \( \boxed{9.59^\circ} \)