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. The possible length of the shortest board, if the third piece is to be at least 5 cm longer than the second, is

• less than 8 cm
• greater than or equal to 8 cm but less than or equal to 22 cm
• less than 22 cm
• greater than 22 cm

To solve the problem, let's define the lengths of the three pieces of board. Let the shortest length be represented as x. According to the information given:

• The second length is x + 3 cm.
• The third length is at least (x + 3) + 5 cm, which simplifies to x + 8 cm.

Now, we can set up the equation for the total length of the board:

x + (x + 3) + (x + 8) = 91 cm.

This simplifies to:

3x + 11 = 91

Subtracting 11 from both sides gives:

3x = 80

Dividing by 3 results in:

x = 80/3 ≈ 26.67 cm.

Now, we need to check the conditions for the possible lengths of the shortest board:

• If x < 8, then x + 3 would be less than 11 cm, and x + 8 would be less than 16 cm, which is not possible since the total must equal 91 cm.
• If 8 ≤ x ≤ 22, this range is valid as it allows for the lengths to add up to 91 cm.
• If x > 22, the total would exceed 91 cm.

Thus, the possible length of the shortest board must be between 8 cm and 22 cm, inclusive.