A man wants to cut three lengths from a single piece of board of length 91 cm.

Question:

A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5 cm longer than the second?

[Hint: If $x$ is the length of the shortest board, then $x,(x+3)$ and $2 x$ are the lengths of the second and third piece, respectively. Thus, $x=(x+3)+2 x \leq 91$ and $2 x \geq(x+3)+5$ ]

Solution:

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

Since the three lengths are to be cut from a single piece of board of length 91 cm,

$x \mathrm{~cm}+(x+3) \mathrm{cm}+2 x \mathrm{~cm} \leq 91 \mathrm{~cm}$

$\Rightarrow 4 x+3 \leq 91$

$\Rightarrow 4 x \leq 91-3$

$\Rightarrow 4 x \leq 88$

$\Rightarrow \frac{4 x}{4} \leq \frac{88}{4}$

$\Rightarrow x \leq 22$   (1)

Also, the third piece is at least 5 cm longer than the second piece.

$\therefore 2 x \geq(x+3)+5$

$\Rightarrow 2 x \geq x+8$

$\Rightarrow x \geq 8 \ldots(2)$

From (1) and (2), we obtain

$8 \leq x \leq 22$

Thus, the possible length of the shortest board is greater than or equal to 8 cm but less than or equal to 22 cm.