Question

A man wants to cut three lengths from a single piece of board of maximum length 91cm. The second length is to be 3 cm longer than the shortest and third length is to be twice as long as the shortest. What are the possible lengths for the shortest board, if third piece is to be atleast 5cm longer than the second? Also, show the possible lengths on a graph paper in Ist quadrant.

Answer 1

Let the length of the shortest piece be x cm. Then the length of the second and third pieces are (x+3) cm and 2x cm, respectively. According to the question,

$x+(x+3)+2x\le 91$ and $2x\ge x+3+5$

$\Rightarrow 4x+3\le 91$ and $2x\ge x+8$

$\Rightarrow 4x+3-3\le 91-3$ and $2x-x\ge x+8-x$

$\Rightarrow 4x\le 88$ and $x\ge 8\Rightarrow \frac{4x}{4}\le \frac{88}{4}$ and $x\ge 8$

$\Rightarrow x\le 22$ and $x\ge 8\Rightarrow 8\le x\le 22$

Hence, the shortest piece must be atleast 8cm long but not more than 22cm long.

Consider the inequalities as equations x=22 and x=8
Here, we know that, lines x=8 and x=22 are perpendicular on X-axis.

Inequality $x\ge 8$ is away from thye origin and $x\le 22$ is towards the origin.

The graph of above inequalities is shown below in 1st quadrant.