The density of air changes with height. Under some conditions density p. depends on height z, and temperature T according to the following equation where Po and À e both constants. A meteorological balloon ascends (i.e., starts at z =1 and gains height) over the course of several hours. Complete parts (a) and (b) below. iz P(z.T) = Po e dz (a) Assuming that the balloon ascends at a speed v (i.e., = v) and that the temperature changes over time (i.e., that T is given by a function T(t)), derive, using the chain rule, an expression for the rate of change of air density, dt as measured by the weather balloon. Choose the correct answer below. dp iz dT O A. = p dt và dp OB. dt dp vÀ iz dT Oc. dt v. dp OD. dt (b) Assume that v = 1, Po = 1, and =1 and that when t=0, T = 1. Are there any conditions under which the density, as measured by the balloon will not change in time? That is, find a differential equation that T must satisfy, if dp = =0, and solve this differential equation. Choose the correct equation for T below. O. T(t) = vt +C O B. T(t) = C(1 + vt) Oc. T(t) = 1+ vt O D. T(t) =vt
The density of air changes with height. Under some conditions density p. depends on height z, and temperature T according to the following equation where Po and À e both constants. A meteorological balloon ascends (i.e., starts at z =1 and gains height) over the course of several hours. Complete parts (a) and (b) below. iz P(z.T) = Po e dz (a) Assuming that the balloon ascends at a speed v (i.e., = v) and that the temperature changes over time (i.e., that T is given by a function T(t)), derive, using the chain rule, an expression for the rate of change of air density, dt as measured by the weather balloon. Choose the correct answer below. dp iz dT O A. = p dt và dp OB. dt dp vÀ iz dT Oc. dt v. dp OD. dt (b) Assume that v = 1, Po = 1, and =1 and that when t=0, T = 1. Are there any conditions under which the density, as measured by the balloon will not change in time? That is, find a differential equation that T must satisfy, if dp = =0, and solve this differential equation. Choose the correct equation for T below. O. T(t) = vt +C O B. T(t) = C(1 + vt) Oc. T(t) = 1+ vt O D. T(t) =vt
Calculus: Early Transcendentals
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Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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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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![## Air Density and Temperature Change with Height
### Overview
The density of air changes with height. Under certain conditions, density \( \rho \) depends on height \( z \) and temperature \( T \) according to the equation:
\[
\rho(z, T) = \rho_0 e^{\frac{\lambda z}{T}}
\]
where \( \rho_0 \) and \( \lambda \) are constants. A meteorological balloon ascends, starting at \( z = 1 \), and gains height over several hours. Address the following problems:
### (a) Rate of Change of Air Density
Assuming the balloon ascends at speed \( v \) (i.e., \( \frac{dz}{dt} = v \)) and temperature changes over time (i.e., \( T \) is a function \( T(t) \)), derive using the chain rule, an expression for the rate of change of air density \( \frac{dp}{dt} \), as measured by a weather balloon. Choose the correct answer:
- **A.** \[ \frac{dp}{dt} = \rho \left(\frac{\lambda z}{T^2} \frac{dT}{dt} \right) \]
- **B.** \[ \frac{dp}{dt} = \rho \left( -\frac{v \lambda}{T} + \frac{\lambda z}{T^2} \frac{dT}{dt} \right) \]
- **C.** \[ \frac{dp}{dt} = \rho \left( -\frac{v \lambda}{T} - \frac{\lambda z}{T^2} \frac{dT}{dt} \right) \]
- **D.** \[ \frac{dp}{dt} = \rho \left( -\frac{v \lambda}{T} - \frac{\lambda z}{T^2} \right) \frac{dT}{dt} \]
### (b) Conditions for Constant Density
Assume \( v = 1 \), \( \rho_0 = 1 \), and \( \lambda = 1 \) with \( t = 0 \), \( T = 1 \). What conditions allow density to remain unchanged over time? Find the differential equation that \( T \) must satisfy if \( \frac{dp](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98f1756b-41d1-4345-9e2e-e05360c283c9%2Fbbaae431-28ab-4e81-b9a4-4ca9c68200dd%2F3mosger_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Air Density and Temperature Change with Height
### Overview
The density of air changes with height. Under certain conditions, density \( \rho \) depends on height \( z \) and temperature \( T \) according to the equation:
\[
\rho(z, T) = \rho_0 e^{\frac{\lambda z}{T}}
\]
where \( \rho_0 \) and \( \lambda \) are constants. A meteorological balloon ascends, starting at \( z = 1 \), and gains height over several hours. Address the following problems:
### (a) Rate of Change of Air Density
Assuming the balloon ascends at speed \( v \) (i.e., \( \frac{dz}{dt} = v \)) and temperature changes over time (i.e., \( T \) is a function \( T(t) \)), derive using the chain rule, an expression for the rate of change of air density \( \frac{dp}{dt} \), as measured by a weather balloon. Choose the correct answer:
- **A.** \[ \frac{dp}{dt} = \rho \left(\frac{\lambda z}{T^2} \frac{dT}{dt} \right) \]
- **B.** \[ \frac{dp}{dt} = \rho \left( -\frac{v \lambda}{T} + \frac{\lambda z}{T^2} \frac{dT}{dt} \right) \]
- **C.** \[ \frac{dp}{dt} = \rho \left( -\frac{v \lambda}{T} - \frac{\lambda z}{T^2} \frac{dT}{dt} \right) \]
- **D.** \[ \frac{dp}{dt} = \rho \left( -\frac{v \lambda}{T} - \frac{\lambda z}{T^2} \right) \frac{dT}{dt} \]
### (b) Conditions for Constant Density
Assume \( v = 1 \), \( \rho_0 = 1 \), and \( \lambda = 1 \) with \( t = 0 \), \( T = 1 \). What conditions allow density to remain unchanged over time? Find the differential equation that \( T \) must satisfy if \( \frac{dp
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