Respuesta :
Answer and Step-by-step explanation:
Solution:
The equation is given by:
z – z0 = a(x – x0) + b(y – y0).
The point (1, 2, 3) is given , substitute them for x0, y0 and z0:
Z – 3 = a (x – 1) + b(y-2)
Both a and b should be negative, because normal vector to point in the negative x and y direction so that plane will enclose a tetrahedron with finite area in the first octant.
This will only happen if z becomes smaller as we move towards the positive x and y directions.
To find expressions for V, firstly find expression for x, y and z intercepts.
X – Intercepts (y and z = 0) : 2b +a -3 / a
y- Intercepts (x, z = 0): 2b + a- 3 / b
z – Intercepts( x, y = 0): 3 – a – 2b
The base is = (2b+a-3)2/ 2ab
Volume is 1/3 base x height, so
V (a, b) = - (2b+a-3)3 / 6ab
The gradient vector:
ΔV (a, b) = |- (3(2b+a-3)2 ab-b (2b+a-3)3)/6a2b2, (6(2b+a-3)2 ab-a(2b+a-3)3)/6a2b2| = 0
b (2b+a-3)2 [ (2b+a-3) -3a] = 0
a (ab + a-3)2 [(2b +a -3) – 6b] = 0
Where a, b not equal to 0.
We cannot solve for 2b +a-3 = 0 because, that will lead to V = 0, which is impossible.
Therefore, we only solve for second halves equations, which are system of linear equations:
-2a + 2b = 3 and a – 4b = 3
Which gives a = -3 and b = -3/2
Put in equation:
Z – 3 = -3 (x-1) – 3/2 (y-2)
3x +3/2 y +z = 9
Multiplying by 2:
6x + 3y + 2z = 18.
Otras preguntas
