```Question 90942
A picture is framed on colored cardboard 40 cm by 32 cm. If there is a uniform boarder around the picture such that the area of the boarder is 720 cm^2, find the dimensions of the picture.
:
Draw a rough diagram of this; label the outer dimension of the frame as 40
by 32.
Label the width of the border as x. It will be apparent that the dimensions
of the picture will be (40-2x) by (32-2x)
:
If we find the width of the border it will be easy to find the dimensions of the picture:
:
Find the area of the colored cardboard: 40 * 32 = 1280 sq/cm
Overall area: 1280
Border area:  -720
-------------------subtract
Picture area: 560 sq/cm
:
A simple area of the picture equation:
(40-2x)(32-2x) = 560
FOIL
1280 - 144x + 4x^2 = 560
:
Arrange as a quadratic equation:
4x^2 - 144x + 1280 - 560 = 0
4x^2 - 144x + 720 = 0
:
Simplify, divide by 4:
x^2 - 36x + 180 = 0
:
Factors to:
(x-6)(x-30) = 0
:
x = +6
and
x = +30
:
Obviously the smaller value is our solution, x = 6
The dimension of the Picture:
40 - 2(6) = 28
32 - 2(6) = 20
;
Check our solution by finding the area
28 * 20 = 560
:
Did this makes sense? Any questions?
```