1. ## Rational roots problem

A box manufacturer makes boxes of volume 500cm3 from pieces of tin 20cm on a side by cutting squares from each corner and folding up the edges. Find the length of each side of a square.

From the above from, I found some steps in solving the above problem. However, I followed through, gotten the answer but couldn't make sense of it. The followings are how I did it:

First : since we know that it cut out squares from the corners, the side of the squares is x. Hence, the length would be 20-2x.

The formula to solve for x would be x(20-x)(20-x). Factoring everything, I will have the answer 5 and 1.9 or 13.09(13.09 is not in the answer sheet)

My questions are, isn't the box suppose to be a square? Why does it have different measurement of the sides 5 and 1.9? Secondly the formula for volume is l*b*h. With sides at 5 and 1.9, I wonder how do i get the volume of 500?

2. ## Re: Rational roots problem

I'm not sure what you mean by "isn't the box supposed to be a square". The box is three dimensional so can't be a square which is two dimensional. Do you mean a cube? It could theoretically be but then 20- 2x would have to be the same as x: 20- 2x= x is the same as 3x= 20 or x= 20/3. Then the volume of the box would be (20/3)^3= 8000/27= 297.29... cubic inches, not 500.

Solving a general cubic can be very difficult but you titled this 'rational roots'. If this equation has a rational root, it must be an integer divisor of the constant term, -50. Trying x= 5 we have 5^3- 20(5^2)+ 100(5)- 125= 125- 500+ 500- 125= 0. Yes, x= 5 is a root. The other two roots are, as you say, irrational numbers around 1.9 and 13.

However, the roots are NOT all lengths of sides. x is a side of the square cut off and the height of the box. Its base is a square with sides of length 20- 2x. Taking x= 5, 20-2x= 20- 10= 10 so the box has a "10 by 10 base" with height 5 so volume 5(10)(10)= 500 as desired. Taking x= 1.9, 20- 2x= 20- 3.8= 16.2. The volume would be 1.9(16.2)(16.2)= 498.636, approximately 500 because we rounded the irrational sides to one decimal place. Taking x= 13 would give base length 20- 26= -6 which is impossible.

There are, in fact, two solutions to this problem: cut 5 cm from each corner to get a 5 by 10 by 10 cm box with volume 500 cc or cut $\displaystyle \frac{15- 5\sqrt{5}}{2}$ cm (approximately 1.9 cm) from each side to get a $\displaystyle \frac{15- 5\sqrt{5}}{2}$ by $\displaystyle 5\sqrt{5}+5$ by $\displaystyle 5\sqrt{5}+ 5$ (approximately 1.9 by 16.2 by 16.2).

3. ## Re: Rational roots problem

Hey hallsofivy, thank you for your reply. But I do not get it on how the height and the base could be of different measurements since a 20x20 sheet is shaved off uniformly in squares. Shouldn't the measurements of height width and base be the same? In addition, you were explaining that height is affected, but the manufacture were cutting off from the sides, which formed a shape of a plus sign. The centre of the plus sign is the base while folded up the edges, they form the height of the box, in that case height isn't affected. Did I get the question wrongly?

4. ## Re: Rational roots problem

Hello, xwy!

A box manufacturer makes boxes of volume 500cm3 from square pieces of tin 20cm on a side
by cutting squares from each corner and folding up the edges. Find the length of each side of a square.

From the above from, I found some steps in solving the above problem.
However, I followed through, gotten the answer but couldn't make sense of it.
The followings are how I did it:

First: since we know that it cut out squares from the corners, the side of the squares is x.
Hence, the length would be 20 - 2x.

The equation solve for x would be: .$\displaystyle x(20 - 2x)(20 - 2x) \:=\:500$

Factoring everything, I will have the answer 5 and 1.9 or 13.09(13.09 is not in the answer sheet)

My questions are, isn't the box suppose to be a square?
Why does it have different measurement of the sides 5 and 1.9?
Secondly the formula for volume is l*w*h.
With sides at 5 and 1.9, I wonder how do i get the volume of 500?

You forgot what $\displaystyle x$ represents.

$\displaystyle x$ = side of the corner squares to be removed.

Solving your cubic equation, we get three roots:
. . [tex]x \:=\:5,

**

5. ## Re: Rational roots problem

Hello, xwy!

A box manufacturer makes boxes of volume 500cm3 from square pieces of tin 20cm on a side
by cutting squares from each corner and folding up the edges. Find the length of each side of a square.

From the above from, I found some steps in solving the above problem.
However, I followed through, gotten the answer but couldn't make sense of it.
The followings are how I did it:

First: since we know that it cut out squares from the corners, the side of the squares is x.
Hence, the length would be 20 - 2x.

The equation solve for x would be: .$\displaystyle x(20 - 2x)(20 - 2x) \:=\:500$

Factoring everything, I will have the answer 5 and 1.9 or 13.09(13.09 is not in the answer sheet)

My questions are, isn't the box suppose to be a square?
Why does it have different measurement of the sides 5 and 1.9?
Secondly the formula for volume is l*w*h.
With sides at 5 and 1.9, I wonder how do i get the volume of 500?

You forgot what $\displaystyle x$ represents.

$\displaystyle x$ = side of the corner squares to be removed.

Solving your cubic equation, we get three roots:
. . $\displaystyle x \:=\:5,\;1.909830056,\;13.09016994$

We have a 20-cm square of tin.
We cannot cut 13-cm squares from each corner, can we?

But we can cut 5-cm squares from each corner.
The resulting box will have a volume of 500 cm3.

The same is true if we cut out 1.909830056-cm squares.