A small craft in Lake Ontario sends out a series of distress signals. The coordinates of the boat in trouble's first signal was (49,68). The coordinates of the second signal, sent one hour later, are (63,80). A rescue boat must be sent to the location of the craft one hour from now. Assuming the second signal is the midpoint between the first signal and the rendezvous ocation, where should the rescue craft be sent?
A small craft in Lake Ontario sends out a series of distress signals. The coordinates of the boat in trouble's first signal was (49,68). The coordinates of the second signal, sent one hour later, are (63,80). A rescue boat must be sent to the location of the craft one hour from now. Assuming the second signal is the midpoint between the first signal and the rendezvous ocation, where should the rescue craft be sent?
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem Statement:**
A small craft in Lake Ontario sends out a series of distress signals. The coordinates of the boat in trouble's first signal was (49,68). The coordinates of the second signal, sent one hour later, are (63,80). A rescue boat must be sent to the location of the craft one hour from now. Assuming the second signal is the midpoint between the first signal and the rendezvous location, where should the rescue craft be sent?
**Solution:**
To find the coordinates where the rescue craft should be sent, we need to determine the rendezvous location. We are told that the second signal is the midpoint between the first signal and the rendezvous location. These are the steps to find the rendezvous coordinates:
1. Let the coordinates of the rendezvous location be (x, y).
2. Using the midpoint formula for coordinates, where (x₁, y₁) and (x₂, y₂) are the endpoints and \( \left(\frac{x₁ + x₂}{2}, \frac{y₁ + y₂}{2}\right) \) is the midpoint:
Given:
- First signal coordinates: (49, 68)
- Second signal (midpoint) coordinates: (63, 80)
\[
\left(\frac{49 + x}{2}, \frac{68 + y}{2}\right) = (63, 80)
\]
3. Solve for x and y:
- For x-coordinate:
\[
\frac{49 + x}{2} = 63
\]
Multiply both sides by 2:
\[
49 + x = 126
\]
Subtract 49 from both sides:
\[
x = 77
\]
- For y-coordinate:
\[
\frac{68 + y}{2} = 80
\]
Multiply both sides by 2:
\[
68 + y = 160
\]
Subtract 68 from both sides:
\[
y = 92
\]
Therefore, the coordinates for the rescue craft to meet the troubled boat are \( (77, 92) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1ec86780-754e-4dc1-88b7-062788875a29%2Fc9c18751-c6c2-428c-95f6-8ab6bfdd9588%2F4ciw4pp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A small craft in Lake Ontario sends out a series of distress signals. The coordinates of the boat in trouble's first signal was (49,68). The coordinates of the second signal, sent one hour later, are (63,80). A rescue boat must be sent to the location of the craft one hour from now. Assuming the second signal is the midpoint between the first signal and the rendezvous location, where should the rescue craft be sent?
**Solution:**
To find the coordinates where the rescue craft should be sent, we need to determine the rendezvous location. We are told that the second signal is the midpoint between the first signal and the rendezvous location. These are the steps to find the rendezvous coordinates:
1. Let the coordinates of the rendezvous location be (x, y).
2. Using the midpoint formula for coordinates, where (x₁, y₁) and (x₂, y₂) are the endpoints and \( \left(\frac{x₁ + x₂}{2}, \frac{y₁ + y₂}{2}\right) \) is the midpoint:
Given:
- First signal coordinates: (49, 68)
- Second signal (midpoint) coordinates: (63, 80)
\[
\left(\frac{49 + x}{2}, \frac{68 + y}{2}\right) = (63, 80)
\]
3. Solve for x and y:
- For x-coordinate:
\[
\frac{49 + x}{2} = 63
\]
Multiply both sides by 2:
\[
49 + x = 126
\]
Subtract 49 from both sides:
\[
x = 77
\]
- For y-coordinate:
\[
\frac{68 + y}{2} = 80
\]
Multiply both sides by 2:
\[
68 + y = 160
\]
Subtract 68 from both sides:
\[
y = 92
\]
Therefore, the coordinates for the rescue craft to meet the troubled boat are \( (77, 92) \).
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