Trace of sphere in the coordinate planes

Question: A sphere is trapped inside the box whose walls are the following six planes:

x = 2; x = 8; y = 0; y = 10; z = 3; z = 13

(a) Find an equation for the sphere, in the form of a level surface of a 3-variable function: g(x; y; z) = C.

(b) Find a function z = f(x; y) whose graph is the bottom half of the sphere.

(c) Describe the trace of the sphere in each of the coordinate planes (possibly the empty set).

(d) Find the point(s) of intersection of the sphere with the line defined by {y = 1, z =2x + 8}

Question: The temperature T (in °C) at any point in the region -10 ≤ x ≤ 10, -10 ≤ y ≤ 10 is given by the function T(x, y) = 100 – x² - y²

(a) Provide f(x, y) and the equation g(x, y, z) = c.

(b) Sketch isothermal curves (curves of constant temperature) for T = 100°C, T = 75°C, T = 50°C, T = 25°
C, and T = 0°C.

(c) A heat-seeking bug is put down at a point on the xy-plane. In which direction should it move to increase its temperature fastest? How is that direction related to the level curve through that point?

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Mathematics: Trace of sphere in the coordinate planes
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