Quadratic equation word problem?

Question

Thomas is going to make an open-top box by cutting equal squares from the four corners of an 11 inch by 14 inch sheet of cardboard and folding up the sides. If the area of the base is to be 80 square inches, then what size square should be cut from each corner? Please explain your answer. Thanks!

Answer 1

I can't draw a diagram, but that would help. So, picture the box and the corners. Label the amount cut off (like the size) from each side x. The new length on each side is going to be 11-2x or 14-2x (as there are two squares on each side). These multiplied together (like a rectangle) form the area of 80. Let's solve.

(11-2x)(14-2x)=80

Expand.

154-50x+4x^2=80

Bring the 80 over

4x^2-50x+74=0

Bring out the two

2(x^2-25x+37)=0

x is about 1.715 solving this equation.

There is actually another solution, somewhere around 10.8. However, this is an unnecessary solution: 10.8*2 is obviously larger than both 11 and 14. We can throw it out.

Check that:

(11-2(1.715))(14-2(1.715))=80

That's your answer.

Hope this helps.

Answer 2

Let x be the side of the squares to be cut out. You are reducing the width and the height by 2x (twice the length of a square) to get the area of the base.

Reduced width * Reduced length = 80 sq. in.
(11 - 2x)(14 - 2x) = 80

Now you can expand out the equation:
154 -28x -22x + 4x² = 80

Reorganize this to a quadratic form:
4x² - 50x + 74 = 0

You can factor out a 2, if you like:
2(2x² - 25x + 37) = 0

Now solve for x (using the quadratic formula)
a = 2
b = -25
c = 37

x = [ -b ± sqrt( b² - 4ac ) ] / 2a
x = [ -(-25) ± sqrt( (-25)² - 4(2)(37) ) ] / 2(2)
x = [ 25 ± sqrt( 625 - 296) ] / 4
x = [ 25 ± sqrt( 329 ) ] / 4

This gives you two answers:
x ≈ 10.7845893
x ≈ 1.71541071

The first answer doesn't make sense because you would be cutting more than the width or length of the original cardboard.

So the answer is that you must cut four squares of dimensions:
1.7154 in. by 1.7154 in. from each corner so that the resulting box has a base of 80 sq. in.

As a double check, the resulting base will have dimensions of:
width = 11 - 2x = 7.56917857
length = 14 - 2x = 10.56917857
width * length = 80 sq. in.

Thus the four squares should each have sides of approximately 1.7154 in.

Answer 3

Start by expressing the area of the base with the squares of equal dimension of x"
(11-2*x)*(14-2*x)=80
multiply it out and solve for x
4x^2-50X+74=0
x=10.785
or
x=1.72
Obviously only the square with side 1.72 can be cut from the cardboard.

j

Answer 4

the 1st ONE Label the 1st consecutive quantity x Label the 2nd x+a million sq. of the 1st = x^2 decreased by 25 = x^2 - 25 Equals three times the 2nd: x^2 - 25 = 3(x +a million)