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Thread: Maximizing Volume

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    Maximizing Volume

    Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes, and one vertex in the plane x+7y+9z=63
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    Quote Originally Posted by UODuck1879 View Post
    Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes, and one vertex in the plane x+7y+9z=63
    Let P(a,b,c) denote the vertex in the plane which produces the largest volume.

    1. V(a,b,c) = a \cdot b \cdot c~\wedge~a+7b+9c=63~\implies~c=\frac19 (63-a-7b)

    Therefore

    V(a,b)=7ab - \frac19 a^2 b -\frac79 ab^2

    2. Differentiate wrt a and wrt b. Both derivatives must be zero:

    \left|\begin{array}{l}7b-\frac29ab - \frac79 b^2=0 \\ \\ 7a - \frac19a^2 - \frac{14}9 ab=0 \end{array} \right.

    3. Solve this system of equations for (a, b). Determine c and afterwards the volume of the box.

    4. I've got V_{max} = 147
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