Answer in Calculus for Tom #263315
A space probe in the shape of the sphere x ^ 2 + y ^ 2 + z ^ 2 = 30 enters Earth’s atmosphere and its surface begins to heat. After 1 hour, the temperature at the point (z. y. 🙂 on the probe’s surface is T(x, y, z) = x – 2y + 5z Find the hottest point on the probe’s surface.
Maximize T(x,y,z)=x-2y+5z
Subject to g(x,y,z)=x2+y2+z2-30=0
“L=T-lambda g”
“=x-2y+5z-lambda(x^2+y^2+z^2-30)”
“L_x=1-2lambda x=0 implies2lambda x=1implies x=frac{1}{2lambda}”
“L_y=-2-2lambda y=0implies 2lambda y=-2implies y=frac{-1}{lambda}”
“L_z=5-2zlambda=0implies 2lambda z=5implies z=frac{5}{2lambda}”
“g(x,y,z)=(frac{1}{2lambda})^2+(frac{-1}{2lambda})^2+(frac{5}{2lambda})^2-30=0”
“frac{1}{4lambda^2}+frac{1}{lambda^2}+frac{25}{4lambda^2}=30”
“frac{1+4+25}{4lambda^2}=30”
“frac{29}{4lambda^2}=30”
“4lambda^2*30=29”
“lambda^2=frac{29}{4*30}” =“frac{29}{120}”
“lambda=sqrt{frac{29}{120}}=” 0.4916
“lambda=0.4916, x=frac{1}{2lambda}=1.0171,y=frac{-1}{lambda}=-2.0342,z=frac{5}{2lambda}=5.0854”
“T(1.0171,-2.0342,5.0854)=(1.0171-2(-2.0342)+5(5.0854)”
=30.5125
Hottest point on the probe surface is (1.0171,-2.0342,5.0854)
