A space probe in the shape of the ellipsoid 4x 2 + y 2 + 4z 2

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A space probe in the shape of the ellipsoid 4x² + y² + 4z² = 16 enters Earth’s atmosphere and its surface begins to heat. After 1 hour, the temperature at the point (x, y, z) on the probe’s surface is T(x, y, z) = 8x² + 4yz - 16z + 600.

Find the hottest point on the probe’s surface.

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To find the hottest point on the surface of the probe we need to maximize the temperature function Tx y z subject to the constraint equation 4x2 y2 4z2 16 which represents the shape of the probe We can use the method of Lagrange multipliers to solve this optimization problem The Lagrangian function is given by Lx y z Tx y z 4x2 y2 4z2 16 Taking the partial derivatives of L with respect to x y z and and setting them equal to zero we get the following system of equations Lx 16x 8x 0 Ly 4z 2y 0 Lz 4y 16 8z 0 L 4x2 y2 4z2 16 0 Solving these equations simultaneously we obtain x 2 y 22 z 2 52 There are eight possible solutions corresponding to the eight points where the probes surface intersects the planes x 2 y 22 and z 2 We need to evaluate the temperature function T ...View the full answer

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