Question

A picture that measures 10 cm by 5 cm is to be surrounded by a matte before being framed. The width of the matte is to be the same on all sides of the picture. The area of the matte is to be twice the area of the picture. What is the width of the matte?

Answer 1

The width of the matte surrounding the picture is 5 cm.

Step-by-step explanation:

The width of the matte can be found by solving the equation for the areas. Let's assume the width of the matte is x cm. The dimensions of the picture including the matte will be (10+2x) cm by (5+2x) cm.

The area of the picture is 10 cm x 5 cm = 50 cm². The area of the matte is twice the area of the picture, so it can be written as 2(10 cm x 5 cm) = 100 cm². Setting up the equation:

100 cm² = (10+2x) cm x (5+2x) cm

Expanding and rearranging the equation:

4x² + 30x + 50 = 0

Solving the quadratic equation gives: x = -5/2 or x = -5. Since the width of the matte cannot be negative, we discard the negative solution. Therefore, the width of the matte is 5 cm.