1. ## Word Problem

Find the volume and dimensions of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 6.

I can't get very far with this problem.

I would really appreciate some help even if your only willing to give a hint.

2. Originally Posted by Undefdisfigure
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 + 2y + 3z = 6.

I can't get very far with this problem.

I would really appreciate some help even if your only willing to give a hint.
did you draw a diagram?

(i assume you're in calculus 3, so getting everything in one variable is not one of my objectives).

draw a diagram to see what i am talking about. the vertex that touches the plane will be the vertex that is opposite the z-axis above the xy-plane. from this we extract the following information.

we define the length to be the side of the base along the x-axis, the width the side of the base along the y-axis and the height to be the height of the vertex that touches the plane.

Let the length of the box be $\displaystyle x$
Let the width of the box be $\displaystyle y$
Then the height of the box is $\displaystyle z = 2 - \frac 13 x - \frac 23 y$ .........(i solved for z from the equation of the plane)

Thus, the volume is given by:

$\displaystyle V = xyz$

$\displaystyle \Rightarrow V = xy \left( 2 - \frac 13x - \frac 23y \right)$

$\displaystyle \Rightarrow V = 2xy - \frac 13x^2y - \frac 23xy^2$

now we want to find the maximum of this function. can you continue? i gave you more than a hint

3. Yes I can continue from there and yes I did draw a diagram. Looks like this is a bit of a tough problem.

I am in Calculus 3 Jhevon. What are you in? abstract algebra (after Calculus 4)?

Thank you very much for the help.

4. Originally Posted by Undefdisfigure
Yes I can continue from there and yes I did draw a diagram. Looks like this is a bit of a tough problem.
not really tough, just tedious. there is a well defined, mechanical method for finding and classifying critical points for functions of two variables, just find the routine in your notes or text and follow the steps. shouldn't be too hard for you

I am in Calculus 3 Jhevon. What are you in? abstract algebra (after Calculus 4)?
nope, i haven't taken abstract algebra yet.

i am in what you may want to think of as calculus 5. it is advanced calculus 2, and is the fourth class after calc 3 (we have two pre-requisites to take before advanced calc 1)