Problem 22 - Calculus Early Transcendentals

Question

Find the points at which the following planes intersect the coordinate axes and find equations of the lines where the planes intersect the coordinate planes. Sketch a graph of the plane. $$-4 x+8 z=16$$

Step-by-Step Solution

Answer

Answer: The points of intersection of the plane with the coordinate axes are (-4, 0, 0) and (0, 0, 2). The equations of the lines where the plane intersects the coordinate planes are x = -4 and z = 2. The plane does not intersect the y-axis and the xz-plane.

Step 1: Find the points where the plane intersects the coordinate axes

(x, y, and z) To find the points where the plane intersects the coordinate axes, we can set the other two coordinates to zero and solve for the remaining coordinate. For the x-axis: $$y = 0$$ and $$z = 0$$ $$-4x + 8(0) = 16$$ $$-4x = 16$$ $$x = -4$$ The point of intersection is $$(-4, 0, 0)$$. For the y-axis: $$x = 0$$ and $$z = 0$$ This plane does not intersect the y-axis because the y-coordinate is not in the equation. For the z-axis: $$x = 0$$ and $$y = 0$$ $$-4(0) + 8z = 16$$ $$8z = 16$$ $$z = 2$$ The point of intersection is $$(0, 0, 2)$$.

Step 2: Find the equations of the lines where the plane intersects the coordinate planes

(xy, yz, and xz) For the intersection of the plane with the xy-plane, set z = 0: $$-4x + 8(0) = 16$$ $$-4x = 16$$ $$x = -4$$ The equation of the line is $$x = -4$$, parallel to the y-axis. For the intersection of the plane with the yz-plane, set x = 0: $$-4(0) + 8z = 16$$ $$8z = 16$$ $$z = 2$$ The equation of the line is $$z = 2$$, parallel to the y-axis. For the intersection of the plane with the xz-plane, the plane does not intersect the xz-plane due to the lack of a y-coordinate in the equation.

Step 3: Sketch a graph of the plane

To sketch the graph, we can plot the points (-4, 0, 0) and (0, 0, 2) and the lines x = -4 and z = 2. The plane is a vertical plane that intersects the x-axis at x = -4 and the z-axis at z = 2. The plane is parallel to the y-axis.