# Find Maximum Volume

• Apr 18th 2010, 08:24 AM
Butum
Find Maximum Volume
Four identical squares are cut from the corners of a rectangular shet of metal with dimensions:

16 meters by 10 meters.

Find the dimensions of the box that has the maximum volume.

V=M^3

Length = 10m
Width = 16m
Height = x

Therefore the Function is:

v=x(16-2x)(10-2x)
v= x(160-12x-4x^2)
v=(160x-12x^2-4x^3)

Find the derivative and solve for v'=0.

v'=160-24x-12x^2

I'm stuck on solving v'=0.

The quadratics discriminant in my calculations is a negative, so using the quadratic formula does not work.

I don't know what the next step is.(Headbang)
• Apr 18th 2010, 08:40 AM
dwsmith
Quote:

Originally Posted by Butum
Four identical squares are cut from the corners of a rectangular shet of metal with dimensions:

16 meters by 10 meters.

Find the dimensions of the box that has the maximum volume.

V=M^3

Length = 10m
Width = 16m
Height = x

Therefore the Function is:

v=x(16-2x)(10-2x)
v= x(160-12x-4x^2)
v=(160x-12x^2-4x^3)

Find the derivative and solve for v'=0.

v'=160-24x-12x^2

I'm stuck on solving v'=0.

The quadratics discriminant in my calculations is a negative, so using the quadratic formula does not work.

I don't know what the next step is.(Headbang)

$\displaystyle \sqrt{24^2-4(-12)(160)} > 0$
• Apr 18th 2010, 08:49 AM
HallsofIvy
Quote:

Originally Posted by Butum
Four identical squares are cut from the corners of a rectangular shet of metal with dimensions:

16 meters by 10 meters.

Find the dimensions of the box that has the maximum volume.

V=M^3

Length = 10m
Width = 16m
Height = x

Therefore the Function is:

v=x(16-2x)(10-2x)
v= x(160-12x-4x^2)

Your middle term is wrong. It should be -32x- 12x= -44x

Quote:

v=(160x-12x^2-4x^3)
v= 160x- 44x^2- 4x^3

Quote:

Find the derivative and solve for v'=0.

v'=160-24x-12x^2
v'= 160- 88x- 12x^2

Quote:

I'm stuck on solving v'=0.

The quadratics discriminant in my calculations is a negative, so using the quadratic formula does not work.

I don't know what the next step is.(Headbang)
With the function corrected, the discriminant is 64. In fact, if you first divide by 4, the equation becomes reasonably easy to factor.