Q

# 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 1/2 m apart, what is the length of the wood required for the rungs?

Q : 3    A ladder has rungs $\small 25$ cm apart. (see Fig. $\small 5.7$). The rungs decrease uniformly in length from $\small 45$ cm at the bottom to $\small 25$ cm at the top. If the top and the bottom rungs are  $\small 2\frac{1}{2}$  m apart, what is  the length of the wood required for the rungs?

[Hint : Number of rungs  $=\frac{250}{25}+1]$

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It is given that
Total distance between top and bottom rung $= 2\frac{1}{2}\ m = 250cm$
Distance between any two rungs = 25 cm
Total number of rungs = $\frac{250}{25}+1= 11$
And it is also given that bottom-most rungs is of 45 cm length and topmost is of 25 cm length.As it is given that the length of rungs decrease uniformly, it will form an AP with $a = 25 , a_{11} = 45 \ and \ n = 11$
Now, we know that
$a_{11}= a+ 10d$

$\Rightarrow 45=25+10d$
$\Rightarrow d = 2$
Now, total  length of the wood required for the rungs is equal to
$S_{11} = \frac{11}{2}\left \{ 2\times 25+(11-1)2 \right \}$
$S_{11} = \frac{11}{2}\left \{ 50+20 \right \}$
$S_{11} = \frac{11}{2}\times 70$
$S_{11} =385 \ cm$
Therefore, the total  length of the wood required for the rungs is equal to  385 cm

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