Suppose that the Celsius temperature at the point (x comma y comma z )on the sphere x squared plus y squared plus z squared equals 1 is Upper T equals 500 xyz squared. Locate the highest and lowest temperatures on the sphere.

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 = ±

From the above calculations,

x, y, z = ±½, ±½, ±

Calculate for the highest temperature:

Tmax = T(½, ½, = 500 * ½ * ½ * ½ =

Calculate for the lowest temperature:

Tmin = T(-½, ½, = 500 * -½ * ½ * ½ =

54

step-by-step explanation: