Answer:
21x +7y +3z = 63
Step-by-step explanation:
The equation of a plane in intercept form is ...
x/a +y/b +z/c = 1
where a, b, c are the x-, y-, and z-intercepts, respectively. The volume in the first octant cut off by this plane will be given by ...
V = (1/6)abc
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The problem is to minimize the volume subject to the constraint that the plane go through the point. That is, our minimization problem is ...
minimize abc/6
subject to 1/a +3/b +7/c = 1
As with many optimization problems, the optimum solution splits the constraint into equal parts: 1/3 +1/3 +1/3 = 1. That is ...
1/a = 3/b = 7/c = 1/3 ⇒ a = 3, b = 9, c = 21
The intercept form equation for the plane is ...
x/3 +y/9 +z/21 = 1
In standard form, this is ...
21x +7y +3z = 63
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Additional comment
The optimization problem can be solved using the method of Lagrange multipliers.
Th Lagrangian is ...
ℒ = (1/6)abc +λ(bc +3ac +7ab -abc)
and the partial derivatives are ...
∂ℒ/∂a = bc/6 +λ(3c +7b -bc) = 0
∂ℒ/∂b = ac/6 +λ(c +7a -ac) = 0
∂ℒ/∂c = ab/6 +λ(b +3a -ab) = 0
∂ℒ/∂λ = bc +3ac +7ab -abc = 0
Solving these simultaneous equations gives (a, b, c, λ) = (3, 9, 21, 1/2).
