Question 3

We will find the number of rungs and then sum their lengths using the arithmetic progression formula.

Step 1 — Calculate the number of rungs

First, let's find the total distance in centimeters. The total distance between the top and bottom rungs is given as $2\frac{1}{2}\text{ m}$.

$= 2.5 \text{ m}$

We convert meters to centimeters. $= 2.5 \times 100 \text{ cm}$

$\boxed{250 \text{ cm}}$

The distance between two successive rungs is 25 cm. Now, we find the number of gaps between the rungs.

$= \frac{\text{Total distance}}{\text{Distance between successive rungs}}$

$= \frac{250}{25}$

$\boxed{10}$

The number of rungs is one more than the number of gaps.

$= \text{Number of gaps} + 1$

$= 10 + 1$

$\boxed{11}$

Step 2 — Define the Arithmetic Progression (AP)

The lengths of the rungs form an Arithmetic Progression. We consider the rungs from the top to the bottom. The length of the top rung is the first term ($a$).

$\boxed{a = 25 \text{ cm}}$

The length of the bottom rung is the last term ($a_n$). There are 11 rungs, so $n = 11$.

$\boxed{a_{11} = 45 \text{ cm}}$

Now, we find the common difference ($d$) of this AP. We use the formula $a_n = a + (n-1)d$.

$a_{11} = a + (11-1)d$

$45 = 25 + 10d$

$45 - 25 = 10d$

$20 = 10d$

$d = \frac{20}{10}$

$\boxed{d = 2 \text{ cm}}$

Step 3 — Calculate the total length of wood

We need to find the sum of the lengths of all 11 rungs. We use the sum formula for an AP: $S_n = \frac{n}{2}(2a + (n-1)d)$. Here, $n = 11$, $a = 25$, and $d = 2$.

$S_{11} = \frac{11}{2}(2(25) + (11-1)2)$

$S_{11} = \frac{11}{2}(50 + 10 \times 2)$

$S_{11} = \frac{11}{2}(50 + 20)$

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

$S_{11} = 11 \times 35$

$\boxed{385 \text{ cm}}$

Answer

The length of the wood required for the rungs is 385 cm.