3. A greenhouse has a glass dome in the shape of the paraboloid z=8-2x²-2y2 and a flat wooden floor at z = 0. Let S be the closed surface formed by the dome and the floor, oriented with outward unit normal. Suppose that the temperature in the greenhouse is given by T(x, y, z) = x²+ y²+3(z− 2)². The temperature gives rise to a heat flux density field F(x, y, z)-kVT - where k is a positive constant that depends on the insulating properties of the medium. Assume that k = 1 on the glass dome and k = 3 on the wooden floor of the greenhouse. (a) Sketch S, clearly labelling any intercepts and the direction of the normal vector. (b) Write down an expression in terms of x, y and z for the vector field F on the greenhouse. (c) By direct calculation (do not use any integral theorems), find the total heat flux F.ndS across the greenhouse in the direction of the outward unit normal.
3. A greenhouse has a glass dome in the shape of the paraboloid z=8-2x²-2y2 and a flat wooden floor at z = 0. Let S be the closed surface formed by the dome and the floor, oriented with outward unit normal. Suppose that the temperature in the greenhouse is given by T(x, y, z) = x²+ y²+3(z− 2)². The temperature gives rise to a heat flux density field F(x, y, z)-kVT - where k is a positive constant that depends on the insulating properties of the medium. Assume that k = 1 on the glass dome and k = 3 on the wooden floor of the greenhouse. (a) Sketch S, clearly labelling any intercepts and the direction of the normal vector. (b) Write down an expression in terms of x, y and z for the vector field F on the greenhouse. (c) By direct calculation (do not use any integral theorems), find the total heat flux F.ndS across the greenhouse in the direction of the outward unit normal.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
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![3. A greenhouse has a glass dome in the shape of the paraboloid
z=8-2x²-2y2
and a flat wooden floor at z = 0. Let S be the closed surface formed by the dome and the floor,
oriented with outward unit normal.
Suppose that the temperature in the greenhouse is given by
T(x, y, z) = x²+ y²+3(z − 2)².
The temperature gives rise to a heat flux density field
F(x, y, z) -kVT
=
where k is a positive constant that depends on the insulating properties of the medium. Assume
that k = 1 on the glass dome and k = 3 on the wooden floor of the greenhouse.
(a) Sketch S, clearly labelling any intercepts and the direction of the normal vector.
(b) Write down an expression in terms of x, y and z for the vector field F on the greenhouse.
(c) By direct calculation (do not use any integral theorems), find the total heat flux
F.ndS
across the greenhouse in the direction of the outward unit normal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff3fb0c3c-aff8-438e-91d7-614ba5054f53%2Fe151e3f3-5384-4027-9b83-e8f18179c842%2Fdg7hp2c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. A greenhouse has a glass dome in the shape of the paraboloid
z=8-2x²-2y2
and a flat wooden floor at z = 0. Let S be the closed surface formed by the dome and the floor,
oriented with outward unit normal.
Suppose that the temperature in the greenhouse is given by
T(x, y, z) = x²+ y²+3(z − 2)².
The temperature gives rise to a heat flux density field
F(x, y, z) -kVT
=
where k is a positive constant that depends on the insulating properties of the medium. Assume
that k = 1 on the glass dome and k = 3 on the wooden floor of the greenhouse.
(a) Sketch S, clearly labelling any intercepts and the direction of the normal vector.
(b) Write down an expression in terms of x, y and z for the vector field F on the greenhouse.
(c) By direct calculation (do not use any integral theorems), find the total heat flux
F.ndS
across the greenhouse in the direction of the outward unit normal.
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Follow-up Question
There was an error in the question, the 'delta T' should not be bold, the equation should read-
F(x,y,z)=−k∇T
The temperature is a scalar function, with that taken into consideration, could you make any necessary changes in the working and answer?
Thank You
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