What is the width of the mat?

Question

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

Answer 1

It will really help you to draw a diagram and label all the parts. 

 

_______________________

|      _{} x                                the width of the mat is x and is the same in the}

|_x____________________    horizontal and vertical dimensions. 

|    |                              |    

|    | 5cm                      |    

| x  |________________  | 

|     _}x            10 cm                          

|______________________

 

Let's figure the area of the mat: 

 

it's 2x + 10 in the horizontal dimension and 

2x + 5 in the vertical dimension. 

 

To find the area we need to multiply these dimensions together and then subtract out the area of the picture in the middle. The problem tells us that this area is twice the picture so we can set it equal to 100. 

 

(2x +10)(2x +5) - 50 = 100

(4x² + 30x + 50) - 50 = 100 

4x² +30x -100 = 0 

factor out a 2

2(2x² + 15x - 50) = 0

2(2x - 5)(x + 10) = 0

used zero product property, and realize that the dimension must be positive, so we can discard the negative answer ( x+10 = 0, x = -10) 

 

2x - 5 = 0 

x = 5/2 or x = 2.5 cm 

 

you can check your answer...... plug it back in. Let me know if this helps.