Question

A picture has the dimension of 8 units by 4 units. A frame of equal width surrounds

Answer 1

Answer: The width of the frame is 3 units

Step-by-step explanation: Please refer to the picture attached for more details (not drawn to scale).

First of all the frame has been described as having equal widths on all sides, so we shall represent the width by x. The picture has been described as having dimensions measuring 8 units by 4 units. That means the area covered by the picture is derived as;

Area = L x W

Area = 8 x 4

Area = 32 square units

However if the area of the frame alone is 108 square units, then the total area when no picture is placed would be 108 square units plus 32 square units. Similarly, the total length of the frame when no picture is placed would be 8 + x + x which equals 8 + 2x. Likewise, the total width of the frame when no picture is placed would be 4 + x + x which gives 4 + 2x. Hence the total area of the frame (without a picture) would be;

Area = L x W

Area = (8 + 2x) * (4 + 2x)

32 + 108 = (8 + 2x) * (4 + 2x)

140 = 32 + 16x + 8x + 4x²

Collect like terms and you now have,

140 - 32 = 4x² + 24x

108 = 4x² + 24x

Rearranging all terms on one side of the equation gives you

4x² + 24x - 108 = 0

Divide all through by 4

x² + 6x - 27 = 0

By factorization, the quadratic expression now becomes,

(x + 9)(x - 3) = 0

**Note that when factorizing, +6 is the sum of the two factors and -27 is the product of the two factors** and the factors are 9 and -3.

Therefore, x + 9 = 0 and x - 3 = 0

When x + 9 = 0 then x = -9

When x - 3 = 0 then x = 3

Knowing that the dimensions cannot be in the negative, we shall choose  x equals 3.

Therefore the width of the frame is 3 units on all sides.