A square picture has an area of 110.25 cm2. To create a border, the picture is centered on a square mat with an area of 331.24 cm2.

Question

Determine the width of mat around the picture.

 

Also, these are the choices to answer the question. Select one:
a. 14.87 cm
b. 1.28 cm
c. 3.85 cm
d. 55.25 cm

 

If I use your solution and it does not match a choice, it is wrong.

Answer 1

Let sides of the picture = x

Let width of mat around picture = w

 

Draw a picture with the square picture of area 110.25 inside the square mat with area of 331.24.

 

These are the equations you get after labeling the picture with the variables.

 

(2w + x)(2w + x) = 331.24

4w² + 4xw + x² = 331.24                        eq1   ----> area of the mat

 

x² = 110.25                                       eq2        ----> area of picture

 

x = √(110.25)

 

 

Substitute eq2 into eq1 to get an equation in terms of w.

 

4w² + 4√(110.25)w + 110.25 = 331.24

 

 

Subtract 331.24 on both sides of the equation.

 

4w² + 4√(110.25)w - 220.99 = 0

 

 

Use the quadratic formula to solve for w.

 

w = (-b ± √(b² - 4ac)) / 2a

 

where:

a = 4

b = 4√(110.25)      ------>   b² = 16(110.25) = 1764

c = -220.99

 

Plug in these values into the formula to find w.  Accept the positive value of w.

Answer 2

x represents the width of the frame

The side of the square is 10.5 cm

The area of the mat is (10.5 + 2x)(10.5 + 2x)

331.24 = (10.5 + 2x)(10.5 + 2x)

 

Can you solve for x?