Arithmetic

Mathematics
1

A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at bottom to 25 cm at top. If the top and the bottom rungs are 2(1/2) apart, what is the length of wood required for rungs.

2

A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at bottom to 25 cm at top. If the top and the bottom rungs are 2(1/2) apart, what is the length of wood required for rungs.

Answer:
Let’s break this problem down step-by-step to find the length of wood required for the rungs of the ladder.

Solution By Steps:

  1. Identify the total height between rungs:

    • The distance between the top and bottom rungs is given as 2.5 meters (which is 250 cm).

  2. Determine the number of rungs:

    • The rungs are 25 cm apart, so the number of gaps between the rungs is:
      \text{Number of gaps} = \frac{250 \text{ cm}}{25 \text{ cm/gap}} = 10 \text{ gaps}
    • Since the rungs start from the bottom, there will be one more rung than the number of gaps, so the number of rungs is:
      \text{Number of rungs} = 10 + 1 = 11

  3. Identify the arithmetic sequence properties:

    • The lengths of the rungs form an arithmetic sequence where the first term (a) is 45 cm, the last term (l) is 25 cm, and the number of terms (n) is 11.

  4. Calculate the common difference (d) in the sequence:

    • Using the formula for the n-th term of an arithmetic sequence: a_n = a + (n-1)d
      • Here, (a_{11} = 25), (a = 45), and (n = 11)
      • Substitute these values to find d:
      25 = 45 + (11-1)d
      25 = 45 + 10d
      10d = 25 - 45
      10d = -20
      d = -2 \text{ cm}

  5. Find the length of wood required using the sum of the arithmetic sequence:

    • The sum of the first n terms of an arithmetic sequence is: (S_n = \frac{n}{2} \left(2a + (n-1)d\right))
      • Here, (n = 11), (a = 45), and (d = -2)
      • Substitute these values to find the sum of the lengths of all rungs:
      S_{11} = \frac{11}{2} \left(2 \times 45 + (11-1) \times (-2)\right)
      S_{11} = \frac{11}{2} \left(90 + 10 \times (-2)\right)
      S_{11} = \frac{11}{2} \left(90 - 20\right)
      S_{11} = \frac{11}{2} \times 70
      S_{11} = \frac{11 \times 70}{2}
      S_{11} = \frac{770}{2}
      S_{11} = 385 \text{ cm}

Final Answer:
The total length of wood required for the rungs is (385) cm.