Answered

IDNLearn.com is the perfect place to get detailed and accurate answers to your questions. Join our Q&A platform to receive prompt and accurate responses from knowledgeable professionals in various fields.

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.

Sagot :

Panoyinidn
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:

Volume of the pyramid = 6 cubic units

Step-by-step 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

We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Accurate answers are just a click away at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.