Problem 24 - 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. $$12 x-9 y+4 z+72=0$$

Step-by-Step Solution

Answer

Answer: The intersection points with the coordinate axes are $$(-6, 0, 0)$$, $$(0, 8, 0)$$, and $$(0, 0, -18)$$. The equations of the lines where the plane intersects the coordinate planes are $$y = \frac{4}{3}x + 8$$ (xy-plane), $$z = \frac{9}{4}y - 18$$ (yz-plane), and $$z = -3x - 18$$ (xz-plane).

Step 1: Find the points at which the plane intersects the coordinate axes

To find the points at which the plane intersects the coordinate axes, set two of the three coordinates equal to zero and solve for the remaining one. 1. Intersection with x-axis (y = 0, z = 0): $$12x + 0 - 0 + 72 = 0$$ Solve for x: $$12x = -72$$ $$x = -6$$ Intersection point: $$(-6, 0, 0)$$ 2. Intersection with y-axis (x = 0, z = 0): $$0 - 9y + 0 + 72 = 0$$ Solve for y: $$-9y = -72$$ $$y = 8$$ Intersection point: $$(0, 8, 0)$$ 3. Intersection with z-axis (x = 0, y = 0): $$0 + 0 + 4z + 72 = 0$$ Solve for z: $$4z = -72$$ $$z = -18$$ Intersection point: $$(0, 0, -18)$$

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

To find the equations of the lines where the plane intersects the coordinate planes, set the coordinate associated with the plane equal to zero and solve for the other two coordinates. 1. Intersection with xy-plane (z = 0): $$12x - 9y + 72 = 0$$ Rearrange the equation to solve for y: $$y = \frac{4}{3}x + 8$$ This is the equation of the line on the xy-plane. 2. Intersection with yz-plane (x = 0): $$-9y + 4z + 72 = 0$$ Rearrange the equation to solve for z: $$z = \frac{9}{4}y - 18$$ This is the equation of the line on the yz-plane. 3. Intersection with xz-plane (y = 0): $$12x + 4z + 72 = 0$$ Rearrange the equation to solve for z: $$z = -3x - 18$$ This is the equation of the line on the xz-plane.

Step 3: Sketch a graph of the plane

Now that we know the intersection points with the coordinate axes and the equations of the lines where the plane intersects the coordinate planes, we can plot these points and lines on a 3D coordinate space. 1. Plot the points at which the plane intersects the coordinate axes: $$(-6, 0, 0)$$, $$(0, 8, 0)$$, and $$(0, 0, -18)$$. 2. Draw the lines connecting the points to show the plane's intersection with the coordinate planes, using the line equations obtained in step 2: $$y = \frac{4}{3}x + 8$$, $$z = \frac{9}{4}y - 18$$, and $$z = -3x - 18$$. 3. Fill in the plane that contains these lines and points to complete the sketch of the plane. This concludes the breakdown of the exercise.