An airliner passes over an airport at noon traveling 500 mi / hr due east. At 1:00 p.m., another airliner passes over the same airport at the same elevation traveling due north at 580 mi / hr. Assuming both airliners maintain their (equal) elevations, how fast is the distance between them changing at 3:00 p.m.? The equation relating the horizontal distance between the first airliner and the airport, a, the horizontal distance between the second airliner and the airport, b, and the horizontal distance between the two airliners, c is Differentiate both sides of the equation with respect to t. dc dt (Do not simplify.) At 3:00 p.m., the distance between the airliners is changing at a rate of about (Round to the nearest tenth as needed.)
An airliner passes over an airport at noon traveling 500 mi / hr due east. At 1:00 p.m., another airliner passes over the same airport at the same elevation traveling due north at 580 mi / hr. Assuming both airliners maintain their (equal) elevations, how fast is the distance between them changing at 3:00 p.m.? The equation relating the horizontal distance between the first airliner and the airport, a, the horizontal distance between the second airliner and the airport, b, and the horizontal distance between the two airliners, c is Differentiate both sides of the equation with respect to t. dc dt (Do not simplify.) At 3:00 p.m., the distance between the airliners is changing at a rate of about (Round to the nearest tenth as needed.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Concept explainers
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
Topic Video
Question
![**Problem Statement:**
An airliner passes over an airport at noon traveling 500 mi/h due east. At 1:00 p.m., another airliner passes over the same airport at the same elevation traveling due north at 580 mi/h. Assuming both airliners maintain their (equal) elevations, how fast is the distance between them changing at 3:00 p.m.?
**Solution Approach:**
1. **Determine the Relationship:**
The equation relating the horizontal distance between the first airliner and the airport, \(a\), the horizontal distance between the second airliner and the airport, \(b\), and the horizontal distance between the two airliners, \(c\) is:
\[
c = \sqrt{a^2 + b^2}
\]
2. **Differentiate with Respect to Time \(t\):**
Differentiate both sides of the equation with respect to \(t\).
\[
\frac{d}{dt}(a^2 + b^2) = 2a \cdot \frac{da}{dt} + 2b \cdot \frac{db}{dt} = 2c \cdot \frac{dc}{dt}
\]
Simplifying gives:
\[
a \cdot \frac{da}{dt} + b \cdot \frac{db}{dt} = c \cdot \frac{dc}{dt}
\]
3. **Calculate Specific Values at 3:00 p.m.:**
- Distance traveled by the first airliner: \( a = 500 \text{ mi/h} \times 3 \text{ hours} = 1500 \text{ miles} \)
- Distance traveled by the second airliner: \( b = 580 \text{ mi/h} \times 2 \text{ hours} = 1160 \text{ miles} \)
- Calculate \( c = \sqrt{1500^2 + 1160^2} \)
4. **Substitute Values and Solve for \(\frac{dc}{dt}\):**
Given the rates \( \frac{da}{dt} = 500 \) mi/h and \( \frac{db}{dt} = 580 \) mi/h, substitute the known values into the differentiated equation to solve for the rate of change of \( c \),](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2dedb04f-ec8e-4d86-87d9-6910be748f13%2F8f4406f5-b11b-429d-81f3-977fec6005fa%2Fie6qa4_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
An airliner passes over an airport at noon traveling 500 mi/h due east. At 1:00 p.m., another airliner passes over the same airport at the same elevation traveling due north at 580 mi/h. Assuming both airliners maintain their (equal) elevations, how fast is the distance between them changing at 3:00 p.m.?
**Solution Approach:**
1. **Determine the Relationship:**
The equation relating the horizontal distance between the first airliner and the airport, \(a\), the horizontal distance between the second airliner and the airport, \(b\), and the horizontal distance between the two airliners, \(c\) is:
\[
c = \sqrt{a^2 + b^2}
\]
2. **Differentiate with Respect to Time \(t\):**
Differentiate both sides of the equation with respect to \(t\).
\[
\frac{d}{dt}(a^2 + b^2) = 2a \cdot \frac{da}{dt} + 2b \cdot \frac{db}{dt} = 2c \cdot \frac{dc}{dt}
\]
Simplifying gives:
\[
a \cdot \frac{da}{dt} + b \cdot \frac{db}{dt} = c \cdot \frac{dc}{dt}
\]
3. **Calculate Specific Values at 3:00 p.m.:**
- Distance traveled by the first airliner: \( a = 500 \text{ mi/h} \times 3 \text{ hours} = 1500 \text{ miles} \)
- Distance traveled by the second airliner: \( b = 580 \text{ mi/h} \times 2 \text{ hours} = 1160 \text{ miles} \)
- Calculate \( c = \sqrt{1500^2 + 1160^2} \)
4. **Substitute Values and Solve for \(\frac{dc}{dt}\):**
Given the rates \( \frac{da}{dt} = 500 \) mi/h and \( \frac{db}{dt} = 580 \) mi/h, substitute the known values into the differentiated equation to solve for the rate of change of \( c \),
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
