Help please! A flat circular plate has the shape of the region x2 + y2 le 1. The

Question

Help please!

A flat circular plate has the shape of the region x2 + y2 le 1. The plate, including the boundary where x2 + y2 = 1, is heated so that the temperature at any point (x,y) is T(x, y) = x2 +2y2 -x. Find the hottest and coldest points on the plate and the temperature at each of these points. Find all relative extremes and saddle points, if any, for f(x, y) = 1/x - 27/y + xy. Show all your work. Include any tests that you may perform. Explain how decided what was happening at each critical point. A space probe in the shape of the ellipsoid 4x2 +y2 + 4z2 = 16 enters the earth's atmosphere and its surface begins to heat. After one hour the temperature at the point (x, y, z) is T(x, y, z) = 8x2+4yz-16z + 600. Find the hottest point on the probe's surface. Use the method of Lagrange multipliers. BONUS: Find the equation of the plane passing through the point (3,2,1) that slices off the region in the first octant with the least volume.

Explanation / Answer

1)

dT/dx = 2x-1 = 0 -> x = 1/2

dT/dy = 4y = 0 -> y = 0

(1/2 , 0) is inside the circle and is a critical point -> T(1/2 , 0) = -0.25 -> minimum point

On boundary:

T(x) = x^2+2y^2-x = x^2 + 2(1-x^2) - x = 2 - x^2 - x -> dT/dx = -2x-1 = 0 -> x = -1/2 -> y = sqrt(3)/2, -sqrt(3)/2

-> T(-1/2 , sqrt(3)/2) = 1/4 + 3/2 + 1/2 = 2.25 -> maximum point

So the hottest points are: (-1/2 , sqrt(3)/2) and (-1/2 , -sqrt(3)/2)

The coldest point is: (1/2 , 0)

2)

f(x,y) = 1/x - 27/y + xy

df/dx = -1/x^2 + y = 0 -> y = 1/x^2

df/dy = 27/y^2 + x = 0 -> x = -27/y^2

Therefore:

x = -27x^4

x(27x^3+1) = 0 -> x = 0 -> not acceptable because y = 1/x^2

x = -1/3 -> y = 9

only critical point is (-1/3 , 9)

d^2f/dx^2 = 2/x^3 = -54

d^2f/dy^2 = -54/y^3 = -2/27

d^2f/dxdy = 1

D = -54 * (-2/27) - 1 = 3 > 0 , d^2f/dx^2 < 0 -> (-1/3 , 9) is a relative minimum.

3)

S(x,y,z) = T(x,y,z) + u(4x^2+y^2+4z^2-16) = 8x^2+4yz-16z+600 + u(4x^2+y^2+4z^2-16)

dS/dx = 16x + 8xu = 0 -> x = 0 or u = -2

dS/dy = 4z+2uy = 0 -> y = -2z/u -> y = -4/(u^2-1)

dS/dz = 4y-16+8uz = 0 -> -8z/u - 16 + 8uz = 0 -> z(u - 1/u) = 2 -> z = 2u/(u^2-1)

dS/du = 0 -> 4x^2+y^2+4z^2-16 = 0

if x = 0 : y^2+4z^2 = 16 -> (16+16u^2)/(u^2-1)^2 = 16 -> 1+u^2 = (u^2-1)^2 -> u = 0 , +sqrt(3) , -sqrt(3)

-> y = 4 , z = 0 and y = -2 , z = sqrt(3) and y = -2 , z = -sqrt(3)

if u = -2 : y = -4/3 , z = -4/3 , x^2 = (16 - 16/9 - 64/9)/4 = 16/9 -> x = -4/3 , 4/3

T(0,4,0) = 600

T(0,-2,sqrt(3)) = 600 - 24sqrt(3)

T(0,-2,-sqrt(3)) = 600 + 24sqrt(3)

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