Q3:

A ladder has rungs 25 cm apart. (see Fig. 5.7). The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and the bottom rungs are $2\frac{1}{2}$ m apart, what is the length of the wood required for the rungs? [Hint : Number of rungs = $\frac{250}{25} + 1$]

Given:

1. The distance between consecutive rungs = $25\text{ cm}$.

2. The length of the bottom rung ($a_1$) = $45\text{ cm}$.

3. The length of the top rung ($a_n$) = $25\text{ cm}$.

4. The total distance between the top and bottom rungs = $2\frac{1}{2}\text{ m} = 250\text{ cm}$.

To Find:

The total length of the wood required for all the rungs, which is the sum of the lengths of all rungs ($S_n$).

Step 1: Determine the number of rungs ($n$)

The total distance between the top and bottom rungs is $250\text{ cm}$. Since the rungs are placed $25\text{ cm}$ apart, the number of intervals is $\frac{250}{25} = 10$.

According to the hint provided, the number of rungs ($n$) is given by:

$n = \frac{\text{Total Distance}}{\text{Distance between rungs}} + 1$

$n = \frac{250}{25} + 1$

$n = 10 + 1 = 11$

[Since there is one more rung than the number of intervals in a sequence of equally spaced items]

Step 2: Identify the Arithmetic Progression (AP) parameters

The lengths of the rungs form an Arithmetic Progression where:

First term ($a$) = $45\text{ cm}$

Last term ($l$ or $a_n$) = $25\text{ cm}$

Number of terms ($n$) = $11$

Step 3: Calculate the sum of the lengths of the rungs

To find the total length of the wood required, we use the formula for the sum of the first $n$ terms of an Arithmetic Progression when the first and last terms are known:

$S_n = \frac{n}{2} (a + l)$

[Where $S_n$ is the sum, $n$ is the number of terms, $a$ is the first term, and $l$ is the last term]

Substituting the known values into the formula:

$S_{11} = \frac{11}{2} (45 + 25)$

$S_{11} = \frac{11}{2} (70)$

$S_{11} = 11 \times 35$

$S_{11} = 385$

Step 4: Conclusion

The total length of the wood required for the rungs is the sum of the lengths of all 11 rungs, which is $385\text{ cm}$.

Final Answer: 385 cm