Homework Help: algebra 2

Posted by herry on Saturday, October 29, 2011 at 6:51pm.

you can make an open box from a piece of flat cardboard. First cut congruent squares from the four corners of the cardboard. Then fold and tape the sides. let x equal the side of each congruent squares as x increases so does the depth of the box the useable area of the cardboard decreases as x increases, and so do the length and width of the box. what happens to the volume of the box? does it increase or deacreas as x increases? would the answer both suprise you? what size square should you cut from the corners to maximize the volume of your box?What are the dimensions of the box in centimeters?

algebra 2 - Damon, Saturday, October 29, 2011 at 7:18pm
length of bottom = L-2x
width of bottom = W -2x

v = x (L-2x)(W-2x)
= x(LW-2Wx-2Lx +4x^2)
= LW x -2Wx^2 -2Lx^2 +4x^3

This is a cubic polynomial and will have maxima and minima. Just looking at it the volume would be huge as x got huge because of the x^3. However x can not be bigger than W/2 in practice so it is not that simple.
Using calculus to find max or min:
dv/dx = LW -4Wx-4Lx + 12 x^2
that is 0 at a max or min
12 x^2 -4(L+W)x + LW = 0
for a given L and W, solve that quadratic for x of min or max
for example for a square sheet of 10 cm on a side

12 x^2 -4(20) + 100 = 0
3 x^2 -20 + 25 = 0
x = [20 +/- sqrt (400 -300) ]/6
x = [20 +/-10]/6
x = 30/6 or 10/6
30/6 is 5 which is half the width so zero volume
so 10/6 is what we have, 1 2/3 deep

Answer this Question

Related Questions

math - An open topped box can be created by cutting congruent squares from each ...
Calculus - A box with an open top is to be made from a square piece of cardboard...
Pre-calc - Congruent squares are cut from the corners of a 20 in. square piece ...
math - An open box is made from a rectangular piece of cardboard, with ...
math - An open-topped box is made from a rectangular piece of cardboard, with ...
Math - An open-topped box can be created by cutting congruent squares from each ...
Math - A piece of cardboard has a length of 60cm and a width of 40cm. In order ...
optimal dimensions - Applications of derivatives You are planning to make an ...
math grade 12 - A open-topped box can be created by cutting congruent squares ...
calculus - You are planning to make an open-top box from an 12 in by 12 in piece...

For Further Reading

First Name:
School Subject:
Answer: