Find the volume of the given solid Q. Q is bounded by x - 4z = -1, x - 4z = 2, -2y + 3z = 3, -2y + 3z = 5, -4y + z = 1, and -4y + z = 2.
D) 6

Question

Quizplus · Accepted Answer

To find the limits of integration, we must first find the intersection points of the planes.

By equating the first two equations, we get: 
x-4z=-1
x-4z=2
-3=0 (this is false, so there is no intersection point in this case)

By equating the next two equations, we get: 
-2y+3z=3
-2y+3z=5
-2=0 (this is false, so there is no intersection point in this case)

By equating the final two equations, we get:
-4y+z=1
-4y+z=2
y=-0.5, z=1.5

Therefore, the limits of integration are:
-0.5 ≤ y ≤ 0
1 ≤ z ≤ 1.5
-4z-1 ≤ x ≤ -4z+2

The volume of the solid can be found by integrating 1 with respect to x, y, and z over the given limits of integration:
∭ Q dV = ∫∫∫ dV = ∫-0.5^0 ∫1^1.5 ∫(-4z-1)^(-4z+2) dx dy dz
= ∫-0.5^0 ∫1^1.5 (6z+3) dy dz = 6

Thus, the answer is B) 11eaa948_cd8d_7251_84bc_ed3b18d1e56d_TB2342_11.

Find the Volume of the Given Solid Q