Answer:
To derive the temperature function T(x, y, z) for the metal ball, we can use the information that temperature is inversely proportional to the distance from the center of the ball, which is the origin (0, 0, 0). Let's first find an expression for T in terms of distance.
(a) Deriving T(x, y, z):
Let r be the distance from the origin to the point (x, y, z). The distance formula in three dimensions is:
r = √(x^2 + y^2 + z^2)
Since T is inversely proportional to r, we can write:
T(x, y, z) ∝ 1/r
To make this proportional relationship explicit, we introduce a constant k:
T(x, y, z) = k/r
We know that at the point (1, 3, 2), T = 100°C. Substituting these values:
100 = k/√(1^2 + 3^2 + 2^2)
100 = k/√(1 + 9 + 4)
100 = k/√14
To solve for k:
k = 100√14
So, the temperature function is:
T(x, y, z) = 100√14 / √(x^2 + y^2 + z^2)
(b) To find the rate of change of T at (1, 3, 2) in the direction toward the point (0, 4, 2), you'll need to find the gradient of T at (1, 3, 2) and then find its component in the direction of (0, 4, 2). The gradient vector ∇T is given by:
∇T = (∂T/∂x, ∂T/∂y, ∂T/∂z)
Calculate the partial derivatives and then evaluate them at (1, 3, 2):
∂T/∂x = (-100√14 * x) / (x^2 + y^2 + z^2)^(3/2)
∂T/∂y = (-100√14 * y) / (x^2 + y^2 + z^2)^(3/2)
∂T/∂z = (-100√14 * z) / (x^2 + y^2 + z^2)^(3/2)
Evaluate these derivatives at (1, 3, 2) and you'll get the gradient ∇T at that point. Then, find the unit vector in the direction of (0, 4, 2), and take the dot product with ∇T to find the rate of change.
(c) To show that at any point in the ball, the direction of greatest increase in temperature points toward the origin, consider the gradient ∇T, which we calculated in part (b). The gradient vector points in the direction of the greatest rate of change of the function T. Since the gradient is given by:
∇T = (-100√14 * x, -100√14 * y, -100√14 * z) / (x^2 + y^2 + z^2)^(3/2)
Notice that the numerator is a scalar multiple of the position vector (x, y, z), and the denominator is the magnitude of the position vector. This means that ∇T is directly proportional to the position vector and points toward the origin (0, 0, 0). Therefore, at any point in the ball, the direction of greatest increase in temperature is indeed given by a vector pointing toward the origin.
Explanation: