Word problem about border width

Question

Elena wants to place a border of uniform width on the bulletin board in her classrom. If the bulletin board measures 42 inches by 74 inches and elena wants to have a useable area of 2448 square inches, what is the widest possible border

Answer 1

let x=the width of the border

Elena loses x inches on both sides and loses x inches top and bottom when she puts up the border.

The dimensions of the useable area, which is a rectangle, are (42-2x) by (74-2x).

(42-2x)(74-2x)=2448

3108-148x-84x+4x²=2448

3108-232x+4x²=2448

3108-2448-232x+4x²=0

4x²-232x+660=0

divide both sides by 4

x²-58x+165=0

165=3*5*11 and 55+3=58

The quadratic factors into...

(x-55)(x-3)=0

x-3=0

x=3 inches for the border

check: 42-3-3=36 and 74-3-3=68

36*68=2448 square inches

 

Answer 2

OK so the new useable area will be the new length times the new width

 

where the new length and new width will be the current values minus some number (x)

 

(42-x)(74-x)=2448        this is the new width * the new length and we know it equals 2448

 

3108 -116x + x² = 2448

 

get all terms to one side of the equation

 

x² -116x + 660 = 0

 

now factor

 

(x-6)(x-110) = 0

 

x - 110 = 0

 

x = 110  (but this can't work since it is bigger than the original lenght and width)

 

x - 6 = 0

 

x = 6  (so the border is equal to 6 in total or 3 on each side)

 

(42-6)  = 36  new width

(74-6)  = 68  new length

 

36 * 68 = 2448

 

Hope this helps