Question

Suppose an object in the shape of 4^2 + ^2

+4z^2=16 enters the upper atmosphere and its surface begins to heat. After 2 hours, the temperature at a point (, , ) is given by
(, , ) = 8^2 + 4 − 16 + 500. Use Lagrange multipliers to find the hottest and coldest points on the surface of the object.

Answer 1

To find the hottest and coldest points on the surface of the object, you can use the method of Lagrange multipliers. This involves setting up an equation which expresses the temperature at a given point in terms of the coordinates of that point, and then solving the equation with the constraint that the points lie on the surface of the object.

In this case, we are looking for the temperature at the point (x,y,z)=(x,y,z) on the surface of the object 4x2 + 4y2 + 4z2 = 16. The temperature at this point is given by T(x,y,z) = 8x2 + 4y2 − 16 + 500.

To solve for the hottest and coldest points, we need to set up the following equation with Lagrange multipliers:

T(x,y,z) = 8x2 + 4y2 − 16 + 500 4x2 + 4y2 + 4z2 = 16

We can then solve this equation for x, y and z to find the hottest and coldest points.