A strip of uniform width is to be cut off of both sides and both ends of a sheet of paper that is 8 inches by 11 inches in order to reduce the size of the paper to an area of 40 square inches.

Question

Find the width of the strip

Answer 1

40 = (8 - 2x)(11 - 2x)

Can you solve for x and answer?

Answer 2

Let the uniform strip width be x. Then the remaining width of the paper when the strip is cut is ( 8 - x )

Let the same strip width be cut of the length of the paper from both sides. Then the remaining length pf the paper is ( 11 - 2x)

The required remaining area = Remaining Length x Remaining width

and 40 in² = ( 8 -x) ( 11 -2x), expand and you get , 40 = 88 -16x -11x + 2x² = 88 -27x + 2x², rearrange

40-88 = 2x² -27 x , -48 = 2x² -27x , and 2x² -27 x + 48 = 0, solve for quadratic equation

Δ = b² - 4ac, b = -27, a = 2 , c = 44 , Δ = ( -27)² - 4 ( 2)(48) = 729 - 384 = 352 since Δ > 0, then the equation has two roots. The roots are:

x = - b ± √Δ / 2a = -(-27) ± √ 345 / 2(2) = 27 ± 18.57 / 4

x1 = 27 + 18.57 / 4 , or x2 = 27 - 18.57 / 4, x1 = 11.4, and x2 = 2.1

now the strip width cannot be 11.4 inches > the maximum dimension of the paper. Then it must be the 2nd value of 1.9 inches to be checked.

2( 2.1)² - 27(2.1) + 48 = 7.22 - 51.3 + 44 = 0.12 Close enough to 0, plug into

( 8 - x) ( 11-2x) = ( 8 -2.1 ) ( 11 - 2(2.1)) = ( 6.1) ( 11 - 4.2 ) = ( 5.9 ) ( 6.8 ) = 40.1 close enough to 40

The strip width to be cut from once from the width of the paper of 8 inches, and twice from the 11 inch length of the paper is 2.1 inches

Answer 3

Assign the variable W to the width of the strip.

After removing the strips, the dimension of the paper will be 8 - 2W by 11 - 2W

We can then write an equation for the new area:

(8 - 2W) * (11 - 2W) = 40

4W2 - 38W + 88 = 40

2W2 - 19W + 24 = 0

Not factorable, so use the quadratic equation to get W = 8 or 1.5

1.5 is the answer we want

Answer 4

A narrow strip of 0.5 inches is cut from both top and bottom of the 11 inch length, making that dimension now 10 inches. Strips measuring 2 inches are cut from both sides of the 8 inch length, making the paper now have the dimensions of 10 x 4, and having an area of 40 sq. inches.

Answer 5

(8 - s)(11 - s) = 40

88 - 8s - 11s + s2 = 40

s2 - 19s + 48 = 0

(s - 16)(s - 3) = 0

s - 16 = 0

s = 16

Well, you can't cut 16" off 8 or 11.

s - 3 = 0

s = 3

The strip has a width of 3 inches.