A bank heist occurs. The robbers flee the bank at 60 km/h due south. As they are leaving, cops that are 20 km east of the bank begin driving west towards the bank at 80 km/h. At what time before the cops reach the bank will they be closest to the robbers?
A bank heist occurs. The robbers flee the bank at 60 km/h due south. As they are leaving, cops that are 20 km east of the bank begin driving west towards the bank at 80 km/h. At what time before the cops reach the bank will they be closest to the robbers?
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 bank heist occurs. The robbers flee the bank at 60 km/h due south. As they are leaving, cops that are 20 km east of the bank begin driving west towards the bank at 80 km/h. At what time before the cops reach the bank will they be closest to the robbers?
**Explanation:**
This problem involves relative motion and can be solved using concepts from calculus and coordinate geometry. The goal is to determine at what point in time the distance between the robbers and the cops is minimized.
1. **Initial Positions:**
- Robbers start at the bank and move south at 60 km/h.
- Cops start 20 km east of the bank and move west toward the bank at 80 km/h.
2. **Trajectory and Speed:**
- The robbers move along the y-axis (south/north direction).
- The cops move along the x-axis (east/west direction).
3. **Objective:**
- Find the time \( t \) (before the cops reach the bank) when the distance between the robbers and the cops is minimized.
**Key Variables:**
- Let \( t \) be the time in hours.
- Robbers' position at time \( t \): \( (0, -60t) \).
- Cops' position at time \( t \): \( (20 - 80t, 0) \).
**Distance Formula:**
The distance \( D(t) \) between the two points can be computed using the distance formula:
\[ D(t) = \sqrt{(20 - 80t)^2 + (0 - (-60t))^2} \]
**Next Steps:**
1. Differentiate \( D(t) \) with respect to \( t \) to find the critical points.
2. Solve \( \frac{dD}{dt} = 0 \) to find the time when the distance is minimized.
3. Verify the solution to ensure it represents a minimum distance scenario.
This setup allows for an application of calculus and analytical geometry to solve a real-world problem involving motion and distance.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F52f1d920-c9ed-486b-bcbe-d0ada595c668%2F379b81a8-9078-4adc-9a2d-0f605ab44881%2Fp12d3ni_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A bank heist occurs. The robbers flee the bank at 60 km/h due south. As they are leaving, cops that are 20 km east of the bank begin driving west towards the bank at 80 km/h. At what time before the cops reach the bank will they be closest to the robbers?
**Explanation:**
This problem involves relative motion and can be solved using concepts from calculus and coordinate geometry. The goal is to determine at what point in time the distance between the robbers and the cops is minimized.
1. **Initial Positions:**
- Robbers start at the bank and move south at 60 km/h.
- Cops start 20 km east of the bank and move west toward the bank at 80 km/h.
2. **Trajectory and Speed:**
- The robbers move along the y-axis (south/north direction).
- The cops move along the x-axis (east/west direction).
3. **Objective:**
- Find the time \( t \) (before the cops reach the bank) when the distance between the robbers and the cops is minimized.
**Key Variables:**
- Let \( t \) be the time in hours.
- Robbers' position at time \( t \): \( (0, -60t) \).
- Cops' position at time \( t \): \( (20 - 80t, 0) \).
**Distance Formula:**
The distance \( D(t) \) between the two points can be computed using the distance formula:
\[ D(t) = \sqrt{(20 - 80t)^2 + (0 - (-60t))^2} \]
**Next Steps:**
1. Differentiate \( D(t) \) with respect to \( t \) to find the critical points.
2. Solve \( \frac{dD}{dt} = 0 \) to find the time when the distance is minimized.
3. Verify the solution to ensure it represents a minimum distance scenario.
This setup allows for an application of calculus and analytical geometry to solve a real-world problem involving motion and distance.
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