A CERTAIN ISOTOPE 14C OF CARBON OCCURS IN LIVING ORGANISMS IN A FIXED PROPORTION TO ORDINARY CARBON. WHEN THAT ORGANISM DIES, ITS 14C DECAYS EXPONENTIALLY, AND ITS HALF-LIFE IS 5,730 YEARS. ASSUME THAT A PIECE OF CHARCOAL FROM A WOOD FIRE WAS FOUND IN THE CAVE CONTAINS ONLY 9% OF 14C EXPECTED IN THE CORRESPONDING PIECE OF WOOD IN A LIVE TREE. HOW LONG AGO WAS THE WOOD BURNED TO FORM THAT CHARCOAL? CHOICES: A. 19,706 YEARS B. 19,606 YEARS C. 19,906 YEARS D. 19,806 YEARS

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Gradient of a Scalar Field

Consider a room or a place where the temperature is denoted by a scalar field T, such that at each point (x, y, z), the temperature is given as T(x, y, z). It is independent of time, in which the temperature rises most quickly. The gradient of T will show the direction at that point by moving away from (x, y, z). How far will the rise in temperature in that direction be determined by the gradient magnitude? Consider a surface that has a height above sea level at point (x, y) which is given by H(x, y). At that point, pointing in the direction of the steepest slope or grade is the gradient of H. The magnitude of the gradient of the vector gives the steepness of the slope at that point. How a scalar field changes in another direction can be measured by the gradient by taking a dot product rather than just the direction of the greatest change?