Question

Help with this question? I got half of it, but can't seem to get the other half. Please explain how you got your answer. Thanks!

1. The equation of a plane in space is x + 3y + 2z = 6.
a.) Accurately sketch the plane on the set of axes, showing all your calculations below.
b.) The three planes of the axes and the plane you have sketched create a triangular pyramid. Find the volume of this pyramid, showing the formula you are using and all
calculations below.

My Answer:
a. I started off by finding my x-intercept. To find my x-intercept I substituted y and z with 0. So: x + 3(0) + 2(0) = 6. After multiplying and adding, I end up with x = 6. So my x-intercept is (6, 0, 0,). T find my y-intercept I substituted x and z with 0. So: 0 + 3y + 2(0) = 6. After multiplying and adding, we come up with 3y = 6. Now we divide each side by 3 to get y = 2. So my y-intercept is (0, 2, 0). To find my z-intercept, I substituted x and y with 0. So, 0 + 3(0) + 2z = 6. After multiplying and adding, I get 2z = 6. Now I divide each side by 2 to get z = 3. So my z-intercpet is (0, 0, 3). After I plotted the points and connected them, they formed a triangular pyramid.

So thats how much i got. I cant seem to figure out how to get the volume. please help.

Answer 1

X - Intercept
x + 3y + 2z = 6
x + 3(0) + 2(0) = 6
x + 0 + 0 = 6
x + 0 = 6
- 0 - 0
x = 6
X - Intercept: (6, 0, 0)

Y - Intercept
x + 3y + 2z = 6
0 + 3y + 2(0) = 6
0 + 3y + 0 = 6
0 + 0 + 3y = 6
0 + 3y = 6
- 0 - 0
3y = 6
3 3
y = 2
Y - Intercept: (0, 2, 0)

Z - Intercept
x + 3y + 2z = 6
0 + 3(0) 2z = 6
0 + 0 + 2z = 6
0 + 2z = 6
- 0 - 0
2z = 6
2 2
z = 3
Z - Intercept: (0, 0, 3)

Volume of the X - Intercept, Y - Intercept, and Z - Intercept
V = ¹/₃(¹/₂lwh)
V = ¹/₃(¹/₂(6)(2)(3))
V = ¹/₃(¹/₂(12)(3))
V = ¹/₃(¹/₂(36))
V = ¹/₃(18)
V = 6 u³

Answer 2

Volume of the pyramid = 6 cubic units

Explanation:

The volume of a triangular pyramid is: V = (1/3)*A*H

where A is the area of the triangle base, and H is the height of the pyramid.

Taking the triangle formed in the x-y plane as the base, its area is computed as follows:

A = (1/2)*6*2 = 6 square units

where 6 and 2 are the measure of the two perpendicular sides of the triangle. This is taken from the x-intercept point (6, 0, 0) and y-intercept point (0, 2, 0).

The height of the pyramid is then the measure of the z-intercept point (0, 0, 3), that is, 3. Replacing in volume formula:

V = (1/3)*A*H

V = (1/3)*6*3

V = 6 cubic units