Question

A rectangular picture measures 6 inches by 8 inches. Simon wants to build a wooden frame for the picture so that the framed picture takes up a maximum area of 100 square inches on his wall. The pieces of wood that he uses to build the frame all have the same width.Write an equation or inequality that could be used to determine the maximum width of the pieces of wood for the frame Simon could create.

Answer 1

`4x^2 + 28x - 52 \leq 0`

Where, x ≥ 0

Explanation:

Given,

The rectangular picture has,

Length = 6 inches,

Width = 8 inches,

Suppose x be the width( in inch ) of the piece of wooden,

After joining the piece,

New length = 6 + 2x ,

New width = 8 + 2x

So, the area of the final figure,

`V=length* width = (6+2x)(8+2x)`

According to the question,

V ≤ 100 inch²

`\implies (6+2x)(8+2x)\leq 100`

`48 + 12x + 16x + 4x^2\leq 100`

`4x^2 + 28x + 48 - 100\leq 0`

`4x^2 + 28x - 52 \leq 0`

But, width can not be negative,

i.e. x ≥ 0