Two planes leave an airport at the same time. Their speeds are 90 miles per hour and 140 miles per hour, and the angle between their courses is 24°. How far apart are they after 1.5 hours? (Round your answer to the nearest whole number.) mi
Two planes leave an airport at the same time. Their speeds are 90 miles per hour and 140 miles per hour, and the angle between their courses is 24°. How far apart are they after 1.5 hours? (Round your answer to the nearest whole number.) mi
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Please help, math trig
![**Problem: Calculating the Distance Between Two Planes**
Two planes leave an airport at the same time. Their speeds are 90 miles per hour and 140 miles per hour, and the angle between their courses is 24°. How far apart are they after 1.5 hours? (Round your answer to the nearest whole number.)
[Input Box] mi
**Explanation:**
This problem involves calculating the distance between two planes using their speeds, the time lapsed, and the angle between their paths. The solution requires the use of trigonometry, specifically the law of cosines, to find the distance after 1.5 hours.
**Calculation Steps:**
1. **Calculate the Distance Each Plane Travels:**
- First Plane: \( \text{Distance} = 90 \, \text{miles/hour} \times 1.5 \, \text{hours} = 135 \, \text{miles} \)
- Second Plane: \( \text{Distance} = 140 \, \text{miles/hour} \times 1.5 \, \text{hours} = 210 \, \text{miles} \)
2. **Apply the Law of Cosines:**
- The formula for the distance (d) between two points given by the law of cosines is:
\[
d = \sqrt{a^2 + b^2 - 2ab \cos(C)}
\]
where \( a = 135 \) miles, \( b = 210 \) miles, and \( C = 24^\circ \).
3. **Substitute the Values and Compute:**
- Convert the angle to radians if necessary.
- Calculate:
\[
d = \sqrt{135^2 + 210^2 - 2 \times 135 \times 210 \times \cos(24^\circ)}
\]
- Compute the distance and round to the nearest whole number.
This problem requires understanding the principles of geometry and trigonometry, making it suitable for advanced high school or college-level mathematics courses.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46cb5339-3b10-44e8-b370-9e828542de7a%2F9839065d-dee1-4628-84e6-d7070e690292%2Fv7vs0h_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem: Calculating the Distance Between Two Planes**
Two planes leave an airport at the same time. Their speeds are 90 miles per hour and 140 miles per hour, and the angle between their courses is 24°. How far apart are they after 1.5 hours? (Round your answer to the nearest whole number.)
[Input Box] mi
**Explanation:**
This problem involves calculating the distance between two planes using their speeds, the time lapsed, and the angle between their paths. The solution requires the use of trigonometry, specifically the law of cosines, to find the distance after 1.5 hours.
**Calculation Steps:**
1. **Calculate the Distance Each Plane Travels:**
- First Plane: \( \text{Distance} = 90 \, \text{miles/hour} \times 1.5 \, \text{hours} = 135 \, \text{miles} \)
- Second Plane: \( \text{Distance} = 140 \, \text{miles/hour} \times 1.5 \, \text{hours} = 210 \, \text{miles} \)
2. **Apply the Law of Cosines:**
- The formula for the distance (d) between two points given by the law of cosines is:
\[
d = \sqrt{a^2 + b^2 - 2ab \cos(C)}
\]
where \( a = 135 \) miles, \( b = 210 \) miles, and \( C = 24^\circ \).
3. **Substitute the Values and Compute:**
- Convert the angle to radians if necessary.
- Calculate:
\[
d = \sqrt{135^2 + 210^2 - 2 \times 135 \times 210 \times \cos(24^\circ)}
\]
- Compute the distance and round to the nearest whole number.
This problem requires understanding the principles of geometry and trigonometry, making it suitable for advanced high school or college-level mathematics courses.
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