Question 26 of 40advanced🔧 ApplyNumerical5 marks

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?

Correct Answer

The possible lengths of the shortest board are from 8 cm to 22 cm, i.e., 8 ≤\leq x ≤\leq 22.

Exercise: EXERCISE 5.1 | Q: 26 | (Chapter: Page 8)
For More Understanding

Explanation

This question from the NCERT textbook requires setting up two inequalities based on the given conditions. The hint provided in the textbook guides students to define x as the shortest length and express the other two lengths in terms of x. The first inequality comes from the total length constraint (sum of three pieces ≤\leq 91 cm), and the second inequality comes from the condition that the third piece must be at least 5 cm longer than the second piece. Solving both inequalities and combining them gives the range of possible values for x.

Solution Steps

  1. Step 1: Let x be the length of the shortest piece. Then the three pieces are: x (shortest), x + 3 (second piece), and 2x (third piece).

  2. Step 2: From the total length condition: x + (x + 3) + 2x ≤\leq 91. Solving: 4x + 3 ≤\leq 91, which gives 4x ≤\leq 88, therefore x ≤\leq 22.

  3. Step 3: From the condition that third piece is at least 5 cm longer than second: 2x ≥\geq (x + 3) + 5. Solving: 2x ≥\geq x + 8, which gives x ≥\geq 8.

  4. Step 4: Combining both inequalities: 8 ≤\leq x ≤\leq 22. Since x represents a length, x must be positive, which is satisfied by x ≥\geq 8.

  5. Step 5: Therefore, the possible lengths of the shortest board range from 8 cm to 22 cm.

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