Respuesta :
Answer:
a) 62.5
b) -62.5
Step-by-step explanation:
Given:
x² + y² + z² = 1 is T = 500xyz²
Required:
Locate the highest and lowest temperatures.
Here,
S = (x² + y² + z² = 1)
T = 500 xyz²
Let's take Lagrange multiplier:
∇T = λ∇S
Thus,
500 yz² = λ(2x)........... 1
500 xz² = λ(2y).............2
500 xy² = λ(2z)..............3
Where, x² = y², 2y² = z²
Therefore,
x²+ y²+z² = 1
=> x² + x² + 2x² = 1
Solve for x:
x = ±½ = y
z = ±[tex] \frac{1}{\sqrt{2}} [/tex]
From the above calculations,
x, y, z = ±½, ±½, ±[tex]\frac{1}{\sqrt{2}} [/tex]
Calculate for the highest temperature:
Tmax = T(½, ½, [tex]\frac{1}{\sqrt{2}}) [/tex] = 500 * ½ * ½ * ½ = [tex] \frac{500}{8} = 62.5 [/tex]
Calculate for the lowest temperature:
Tmin = T(-½, ½, [tex]\frac{1}{\sqrt{2}}) [/tex] = 500 * -½ * ½ * ½ = [tex] - \frac{500}{8} = -62.5[/tex]
The highest temperatures is 62.5 and lowest temperatures is -62.5 on the sphere
Given-
The given equation is,
[tex]x^2+y^2+z^2=1[/tex]
The equation for the temperature is,
[tex]T=500(xyz)^2[/tex]
By the Lagrange multiplier
[tex]\bigtriangledown T=\lambda \bigtriangledown S[/tex]
we can rewrite the temperature equation by the above multiplier as,
[tex]500yz^2=\lambda (2x)[/tex]
[tex]500xz^2=\lambda (2y)[/tex]
[tex]500xy^2=\lambda (2z)[/tex]
Here let,
[tex]x^2=y^2[/tex]
[tex]2y^2=z^2[/tex]
therefore,
[tex]2x^2=z^2[/tex]
Put this values in the given equation of the question we get,
[tex]x^2+x^2+2x^2=1[/tex]
[tex]4x^2=1[/tex]
[tex]x^2=\dfrac{1}{4}[/tex]
[tex]x= \pm\dfrac{1}{2}[/tex]
Therefore,
[tex]y= \pm\dfrac{1}{2}[/tex]
[tex]z= \pm\dfrac{1}{\sqrt{2} }[/tex]
Use positive value for the highest temperatures and negative values for the lowest value of temperatures in the temperature equation.
Highest temperature is,
[tex]T=500(xyz)^2[/tex]
[tex]T_h=500(\dfrac{1}{2} \times\dfrac{1}{2}\times\dfrac{1}{2} )[/tex]
[tex]T_h=\dfrac{500}{8}=62.5[/tex]
Lowest temperature is,
[tex]T_l=500(\dfrac{-1}{2} \times\dfrac{-1}{2}\times\dfrac{-1}{2} )[/tex]
[tex]T_l=-62.5[/tex]
Hence, the highest temperatures is 62.5 and lowest temperatures is -62.5 on the sphere
For more about the temperatures follow the link given below-
https://brainly.com/question/15267055
