1. ## 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

2. Originally Posted by UODuck1879
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 $V_{max} = 147$