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Question 1:Atmospheric pressure P decreases as altitude h increases. At a temperature of 15°C, the pressure is 101.3 kilopascals (kPa) at sea level, 87.1 kPa at h = 1 km, and 74.9 kPa at h = 2 km. Use a linear approximation to estimate the atmospheric pressure at an altitude of 3 km. Please help me explain how to solve these word problems! From the pictures and the word problem above, thanks

Question 1:Atmospheric pressure P decreases as altitude h increases. At a temperature of 15°C, the pressure is 101.3 kilopascals (kPa) at sea level, 87.1 kPa at h = 1 km, and 74.9 kPa at h = 2 km. Use a linear approximation to estimate the atmospheric pressure at an altitude of 3 km. Please help me explain how to solve these word problems! From the pictures and the word problem above, thanks

Calculus: Early Transcendentals
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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Question 1:Atmospheric pressure P decreases as altitude h increases. At a temperature of 15°C, the pressure is 101.3 kilopascals (kPa) at sea level, 87.1 kPa at h = 1 km, and 74.9 kPa at h = 2 km. Use a linear approximation to estimate the atmospheric pressure at an altitude of 3 km. Please help me explain how to solve these word problems! From the pictures and the word problem above, thanks!
### Related Rates Problems - Educational Examples #### Q.5 A lighthouse is located on a small island 4 km away from the nearest point \(P\) on a straight shoreline. Its beacon of light completes one full revolution every 10 seconds. The task is to determine the speed at which the beam of light is moving along the shoreline when the beam is 1 km away from point \(P\). #### Q.6 A streetlight is set up at the top of a vertical pole that is 36 ft tall. There is a flag being raised at a speed of 3 ft/sec on a vertical flagpole situated 45 ft away from this light pole. The goal is to find out how fast the shadow of the flag is moving along the ground when the flag is 9 ft above the ground.
Transcribed Image Text:### Related Rates Problems - Educational Examples #### Q.5 A lighthouse is located on a small island 4 km away from the nearest point \(P\) on a straight shoreline. Its beacon of light completes one full revolution every 10 seconds. The task is to determine the speed at which the beam of light is moving along the shoreline when the beam is 1 km away from point \(P\). #### Q.6 A streetlight is set up at the top of a vertical pole that is 36 ft tall. There is a flag being raised at a speed of 3 ft/sec on a vertical flagpole situated 45 ft away from this light pole. The goal is to find out how fast the shadow of the flag is moving along the ground when the flag is 9 ft above the ground.
**Problem Description: Baseball Player and Catcher Scenario** *Objective:* Solve the following three problems. Each solution submitted must include: - A diagram on which all variables and constants are clearly labeled. - A precise statement of the premises (i.e., the value of each specified derivative). - An equation relating your variables. - All details leading to the final derivative (rate) equation and all computations associated with your final answer. *Problem Statement:* A baseball player can run at a top speed of 24 ft/sec. A catcher can throw a ball at a speed of 120 ft/sec. When the player attempts to steal third base, the catcher (who is 90 feet from third base) throws the ball toward the third baseman when the runner is 30 feet from third base and running at top speed. Question: At what rate is the distance between the runner and the ball changing exactly 1/2 second after the catcher releases the ball? **Solution Steps:** 1. **Diagram Creation:** - Draw a diagram representing the baseball field situation. Label the positions of the player, catcher, and third base. - Indicate the distances: 90 feet (catcher to third base), 30 feet (runner from third base), and the positions of the runner and the ball. 2. **Premises Identification:** - Speed of the baseball player: 24 ft/sec. - Speed of the ball: 120 ft/sec. - Initial distance of the runner from third base: 30 feet. - Initial distance from the catcher to third base: 90 feet. 3. **Relation Equation:** - Let: - \( x \) be the position of the runner relative to third base. - \( y \) be the position of the ball relative to third base. - Form an equation relating \( x \) and \( y \) over time \( t \). 4. **Rate of Change Calculation:** - Use derivative equations to calculate the rate at which the distance between the ball and the runner is changing 1/2 second after the ball is thrown. 5. **Detailed Steps and Computations:** - Start from the given premisses and derive the equation and then calculate the rate. By working through the steps methodically, you’ll be able to find the rate at which the distance between the runner and the ball is changing after half a second.
Transcribed Image Text:**Problem Description: Baseball Player and Catcher Scenario** *Objective:* Solve the following three problems. Each solution submitted must include: - A diagram on which all variables and constants are clearly labeled. - A precise statement of the premises (i.e., the value of each specified derivative). - An equation relating your variables. - All details leading to the final derivative (rate) equation and all computations associated with your final answer. *Problem Statement:* A baseball player can run at a top speed of 24 ft/sec. A catcher can throw a ball at a speed of 120 ft/sec. When the player attempts to steal third base, the catcher (who is 90 feet from third base) throws the ball toward the third baseman when the runner is 30 feet from third base and running at top speed. Question: At what rate is the distance between the runner and the ball changing exactly 1/2 second after the catcher releases the ball? **Solution Steps:** 1. **Diagram Creation:** - Draw a diagram representing the baseball field situation. Label the positions of the player, catcher, and third base. - Indicate the distances: 90 feet (catcher to third base), 30 feet (runner from third base), and the positions of the runner and the ball. 2. **Premises Identification:** - Speed of the baseball player: 24 ft/sec. - Speed of the ball: 120 ft/sec. - Initial distance of the runner from third base: 30 feet. - Initial distance from the catcher to third base: 90 feet. 3. **Relation Equation:** - Let: - \( x \) be the position of the runner relative to third base. - \( y \) be the position of the ball relative to third base. - Form an equation relating \( x \) and \( y \) over time \( t \). 4. **Rate of Change Calculation:** - Use derivative equations to calculate the rate at which the distance between the ball and the runner is changing 1/2 second after the ball is thrown. 5. **Detailed Steps and Computations:** - Start from the given premisses and derive the equation and then calculate the rate. By working through the steps methodically, you’ll be able to find the rate at which the distance between the runner and the ball is changing after half a second.
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