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Light cones The idea of a light cone appears in the Special Theory of Relativity. The $x y$ -plane (see figure) represents all of three-dimensional space, and the $z$ -axis is the time axis ( $t$ -axis). If an event $E$ occurs at the origin, the interior of the future light cone $(t>0)$ represents all events in the future that could be affected by $E,$ assuming that no signal travels faster than the speed of light. The interior of the past light cone $(t<0)$ represents all events in the past that could have affected $E,$ again assuming that no signal travels faster than the speed of light.a. If time is measured in seconds and distance $(x \text { and } y)$ is measured in light-seconds (the distance light travels in $1 \mathrm{s}$ ), the light cone makes a $45^{\circ}$ angle with the $x y$ -plane. Write the equation of the light cone in this case.b. Suppose distance is measured in meters and time is measured in seconds. Write the equation of the light cone in this case given that the speed of light is $3 \times 10^{8} \mathrm{m} / \mathrm{s}$(FIGURE CAN'T COPY)

a. $x^{2}+y^{2}+z^{2}=c^{2} t^{2}$b. $x^{2}+y^{2}+z^{2}=v_{1}^{2} t^{2}, \quad v_{1}<c$

Calculus 1 / AB

Calculus 3

Chapter 13

Functions of Several Variables

Section 1

Planes and Surfaces

Functions

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