Problem

JQ2009 · #1 $Old$ November 3rd, 2009, 12:43 PM

I have this question which I have no idea how to even start. Its part of a task on differentiation so all I know is that it requires differentiation at some point but I have no clue how to start it.

A rectangular box has a square base and no top. The total area of its 5 sides is 300cm^2. I have to work out the largest possible volume of the box and the dimensions of the box with this volume.

Anyone any ideas?

Keithfert488 · #2 $Old$ November 3rd, 2009, 02:28 PM

Well, you know that $4xy+x^2=300$ because the surface area is 300.

If you derive that equation using implicit differentiation with respect to x, you will find that $dy/dx=-x/2$

You also know that $V=x^2y$

Using implicit differentiation again with respect to x, you find that $dv/dx=x^2(dy/dx)+2xy$

Plugging in $dy/dx=-x/2$ , you find that $dv/dx=(-x^3/2)+2xy$ . You are trying to find when $dv/dx$ is going from positive to negative. So you set it to 0. Now you have a system of equations involving x and y.

$4xy+x^2=300$
$2xy-(x^3/2)=0$

You can solve for y in the top equation and you will find $y=(300-x^2)/(4x)$ . Plug that into the second equation and you will get $0=(300-x^2)/2-(x^3/2)$ .

Multiply by -2 and you get $0=x^3+x^2-300$ . Solving this yields $x=6.377$ . Plug that into the surface area equation to find y. You find that $y=10.167$ . Then, plug both of those values into $V=x^2y$ . You find that $V=413.443 cm^3$