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 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 at least $5$cm longer than the second?

• A. $3\le x\le 91$
• B. $5\le x\le 91$
• C. $3\le x\le 5$
• D. $8\le x\le 22$

Answer 1

The correct option is D $8\le x\le 22$

Let $x$ be the length of the shortest piece.

Then length of the second price is $(x+3)$ cm.

And length of the third price is $2x$ cm.

Total length of the three prices must be $\le 91$ cm.

$x+(x+3)+2x\le 91$

$\Rightarrow 4x+3\le 91$

Subtracting $3$ on both sides, we get

$4x+3-3\le 91-3$

$\Rightarrow 4x\le 88$

Dividing both sides by $4,$ we get

$x\le 22$ .............(1)

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

$2x\ge (x+3)+5$

$\Rightarrow 2x\ge x+8$

Subtracting $x$ from both sides,

$2x-x\ge 8+x-x$

$x\ge 8$ ...............(2)

From (1) and (2)

$8\le x\le 22$

The possible length of the shortest side must be greater than or equal to 8 cm and less than or equal to 22 cm.