Question

A picture that measures 8 cm by 6 cm is to be surrounded by a mat. The mat is to

Answer 1

Answer:

1.4 inches

Step-by-step explanation:

The picture is a rectangle 8 cm by 6 cm. The area of a rectangle is length * width. The area of the picture is 8 cm * 6 cm = 48 cm^2

After the mat is applied, the area doubles, so the new area will be 2 * 48 cm^2 = 96 cm^2.

Let the width of the mat be x. The mat has the same width all around the rectangular picture, so it adds x on each side of the length and x on each side of the width.

old length: 8

new length: 2x + 8

old width: 6

new width: 2x + 6

Area of the new rectangle with mat = new length * new width

area = (2x + 8)(2x + 6)

The new area is 96, so that give us an equation.

(2x + 8)(2x + 6) = 96

Use FOIL on the left side:

4x^2 + 12x + 16x + 48 = 96

Combine like terms, and subtract 96 from both sides:

4x^2 + 28x - 48 = 0

Divide both sides by 4:

x^2 + 7x - 12 = 0

To factor the trinomial, we need two numbers that add to 7 and multiply to -12. There are no such numbers, so we need to use the quadratic formula.

x = [-b +/- sqrt(b^2 - 4ac)]/(2a)

x = [-7 +/- sqrt(7^2 - 4(1)(-12)]/[2(1)]

x = [-7 +/- sqrt(49 + 48)]/2

x = [-7 +/- sqrt(97)]/2

x = 1.4 or x = -8.4

Since the mat cannot have a negative width, the negative solution is discarded.

Answer: The width of the mat is 1.4 inches.