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Custom Made 16" Used Cutting Saddle- (Cutter Western Horse Tack) $875.00

Hand Made Cutting Saddle 16 1/2 Inch seat. Great seat very well made. $900.00

New Silver Oaks Cutting/Roping Saddle by Easy Rider 17" Made in the USA $1,370.00

An open box is to be made from a decorative scrapbooking card, 40 cm by 30 cm, by cutting squares from the?

An open box is to be made from a decorative scrapbooking card, 40 cm by 30
cm, by cutting squares from the corners. Find the height of the box that has the maximum volume.

Ok, I will help you with the setup here....I assume you can do the basics of the calculus involved (derivatives and such).

Ok, for this to make sense, its best if you draw a picture as you are following along...
First, draw your rectangle with dimensions labeled 40 x 30.

Now, when we cut the corners out, this is where the seems will be when we fold the sides up to make the box. It should also be noted that every corner must be cut identical.

Ok, the unknown in this problem is the "height" of the box which we will call "x"
Note: "x" will occur twice on each side of your current rectangle.

If you were to look at the sides of your rectangle, you will notice that the dimensions can no longer be 40 x 30, because we cut a length of "x" off of each corner. When we rewrite the dimensions to account for this, you should notice that both sides are 2(x) shorter than they were originally. This is how we relate all of our sides.

When you fold the box up, you shouldd recognize that its dimensions are now (40-2x) X (30-2x) X "x"
From here, the problem is child's play. The volume of a box is L*W*H, so your volume in terms of "x" is 1200x - 140x^2 + 4x^3

Recall from your classes: maximums and miniums occur when the first derivative = 0. Also note, we only care about positive values for x, since we cannot have negative length.

1st derivative = 1200-280x+12x^2
set that equal to 0 or find the solution graphically on a graphing calculator. My guess is that it will not factor nicely and you will have to use quadratic formula or find zeros by graphing.

I dont have my graphing calc handy right now, but I assume you can take it from here...if not, you have bigger issues than this type of question..

Note: you can also skip the whole first derivative phase if you understand the max and min...you can go directly to the graph of 1200x-140x^2+4x^3 and notice you will have 2 separate scenarios that will produce a max or min.

Good luck.

Custom Made 16" Used Cutting Saddle- (Cutter Western Horse Tack) $875.00

Hand Made Cutting Saddle 16 1/2 Inch seat. Great seat very well made. $900.00

New Silver Oaks Cutting/Roping Saddle by Easy Rider 17" Made in the USA $1,370.00

Home made Gear cutting Hobs part3

An open box is to be made by cutting out squares from the corners of a rectangular piece of cardboard and then

turning up the sides. If the piece of cardboard is 10cm by 22 cm, what are the dimensions of the box of largest volume made in this way?

Let x = length of the square you are cutting out. This will also be the height of box

When you cut x off of each corner of the 10 cm side then that will now be 10 -2x

The same goes for the 22 cm side it will be 22 -2x

length = 10-2x
width = 22-2x
height=x

V=lwh
V= x(10-2x)(22-2x)
V=220x -64x^2 +4x^3 now find the derivative
V'= 220 -128x +12x^2 set = to 0 and solve
x= 8.5 or x=2.2

x cannot = 8.5 because 2x= 17 and there would be nothing left of the 10cm side

so h= 2.2cm
l = 5.6cm
w = 17.6 cm