A hot-air balloon is held at a constant altitude by two ropes that are anchored on the ground. One rope is 120 ft long and makes an angle of 65 with the ground. The other rope is 115 ft long. What is the distance between the points on the ground at which the two ropes are anchored? Round your answers to the nearest tenth of a foot. 88.1 feet only 88.1 feet or 13.3 feet 170.5 feet No solution is possible
A hot-air balloon is held at a constant altitude by two ropes that are anchored on the ground. One rope is 120 ft long and makes an angle of 65 with the ground. The other rope is 115 ft long. What is the distance between the points on the ground at which the two ropes are anchored? Round your answers to the nearest tenth of a foot. 88.1 feet only 88.1 feet or 13.3 feet 170.5 feet No solution is possible
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section: Chapter Questions
Problem 75RE
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![**Homework Help: Geometry and Trigonometry Concepts**
**Problem Statement:**
A hot-air balloon is held at a constant altitude by two ropes that are anchored on the ground. One rope is 120 feet long and makes an angle of 65° with the ground. The other rope is 115 feet long. What is the distance between the points on the ground at which the two ropes are anchored? Round your answer to the nearest tenth of a foot.
**Options:**
- a) 88.1 feet only
- b) 88.1 feet or 13.3 feet
- c) 170.5 feet
- d) No solution is possible
**Explanation and Solution:**
Let’s label the given information:
- Rope 1: 120 feet, angle with ground: 65°
- Rope 2: 115 feet
Since the hot-air balloon is held at a constant altitude, both ropes end at the same elevation above the ground. To determine the horizontal distance between the points on the ground where the ropes are anchored, we can use trigonometric identities and the properties of triangles.
1. **Calculate the height (h) at which the balloon is flying:**
Using the first rope (120 feet, angle 65°):
\[ h = 120 \sin(65°) \]
2. **Calculate the distance on the ground from the balloon directly down to the point where Rope 1 is anchored (b1):**
Using the first rope (120 feet, angle 65°):
\[ b1 = 120 \cos(65°) \]
3. **Verify the height using the second rope's distance:**
Using the second rope (115 feet):
\[ h = 115 \cos(\theta_2) \]
(where \(\theta_2\) is the angle the second rope makes with the ground.)
Now, we've determined h from step 1.
4. **Calculate \(\theta_2\):**
\[ \cos(\theta_2) = \frac{h}{115} \]
Next, calculate the ground distance from the balloon directly down to the point where Rope 2 is anchored (b2);
\[ b2 = 115 \sin(\theta_2) \]
5. **Finally, determine the distance between both points on the ground (d):**
\[ d = b1 + b2 \]
Verify each step, calculate the angles and distances,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Febfa71ca-f003-4e76-b16d-d56ed3281ebe%2F1971d9fe-5044-475d-b7e9-7f050fa582c3%2Feok3o2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Homework Help: Geometry and Trigonometry Concepts**
**Problem Statement:**
A hot-air balloon is held at a constant altitude by two ropes that are anchored on the ground. One rope is 120 feet long and makes an angle of 65° with the ground. The other rope is 115 feet long. What is the distance between the points on the ground at which the two ropes are anchored? Round your answer to the nearest tenth of a foot.
**Options:**
- a) 88.1 feet only
- b) 88.1 feet or 13.3 feet
- c) 170.5 feet
- d) No solution is possible
**Explanation and Solution:**
Let’s label the given information:
- Rope 1: 120 feet, angle with ground: 65°
- Rope 2: 115 feet
Since the hot-air balloon is held at a constant altitude, both ropes end at the same elevation above the ground. To determine the horizontal distance between the points on the ground where the ropes are anchored, we can use trigonometric identities and the properties of triangles.
1. **Calculate the height (h) at which the balloon is flying:**
Using the first rope (120 feet, angle 65°):
\[ h = 120 \sin(65°) \]
2. **Calculate the distance on the ground from the balloon directly down to the point where Rope 1 is anchored (b1):**
Using the first rope (120 feet, angle 65°):
\[ b1 = 120 \cos(65°) \]
3. **Verify the height using the second rope's distance:**
Using the second rope (115 feet):
\[ h = 115 \cos(\theta_2) \]
(where \(\theta_2\) is the angle the second rope makes with the ground.)
Now, we've determined h from step 1.
4. **Calculate \(\theta_2\):**
\[ \cos(\theta_2) = \frac{h}{115} \]
Next, calculate the ground distance from the balloon directly down to the point where Rope 2 is anchored (b2);
\[ b2 = 115 \sin(\theta_2) \]
5. **Finally, determine the distance between both points on the ground (d):**
\[ d = b1 + b2 \]
Verify each step, calculate the angles and distances,
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