Answer

Verified

411.6k+ views

Hint: -Length of the rungs decrease uniformly therefore they will form an A.P.

According to the question it is given that the rungs are 25cm apart and the top and bottom rungs are ${\text{2}}\dfrac{1}{2}{\text{m}}$apart.

$ \Rightarrow {\text{2}}\dfrac{1}{2}m = \dfrac{5}{2}m = \dfrac{{500}}{2}cm = 250cm.\left[ {\because {\text{ 1m = 100cm}}} \right]$

$\therefore $Total number of rungs${\text{ = }}\dfrac{{{\text{Distance between total rungs}}}}{{{\text{Distance between two rungs}}}} + 1$, (plus one because of the bottom rung and thereafter all rungs are 25 cm apart).

$\therefore $Total number of rungs${\text{ = }}\dfrac{{250}}{{25}} + 1 = 11$

Now as the length of the rungs decrease uniformly therefore they will form an A.P.

So the A.P becomes $\left( {45,.................,25} \right)$

So, first term $\left( {{a_1}} \right)$of an A.P${\text{ = 45}}$, last term $\left( {{a_n}} \right)$of an A.P${\text{ = 25}}$, and number of terms in this A.P$ = 11$

Now, as we know last term of this series is written as

${a_n} = {a_1} + \left( {n - 1} \right)d$, Where d is the common difference.

$ \Rightarrow d = \dfrac{{{a_n} - {a_1}}}{n} + 1 = \dfrac{{25 - 45}}{{11}} + 1 = \dfrac{{ - 20}}{{11}} + 1 = \dfrac{{ - 9}}{{11}}$

So, the length of the rungs decrease uniformly by $\dfrac{{ - 9}}{{11}}cm$

The length of the wood required for the rungs equals the sum of all the terms of this A.P

${S_n} = 45 + \left( {45 - \dfrac{9}{{11}}} \right) + \left( {45 - \dfrac{9}{{11}} - \dfrac{9}{{11}}} \right) + .............. + 25$

Therefore sum of this A.P${\text{ = }}{{\text{S}}_n} = \dfrac{n}{2}\left( {{a_1} + {a_l}} \right)$

$ \Rightarrow {{\text{S}}_n} = \dfrac{{11}}{2}\left( {45 + 25} \right) = 11 \times 35 = 385cm$

Therefore the length of the wood required for the rungs${\text{ = 385cm}}$.

Note: -In such types of questions first find out the total numbers of rungs, then the key concept is that the length of the rungs decrease uniformly so, they will form an A.P so, the length of the wood required for the rungs equals the sum of all the terms of this A.P, so apply the formula of sum of an A.P which is stated above, we will get the required answer.

According to the question it is given that the rungs are 25cm apart and the top and bottom rungs are ${\text{2}}\dfrac{1}{2}{\text{m}}$apart.

$ \Rightarrow {\text{2}}\dfrac{1}{2}m = \dfrac{5}{2}m = \dfrac{{500}}{2}cm = 250cm.\left[ {\because {\text{ 1m = 100cm}}} \right]$

$\therefore $Total number of rungs${\text{ = }}\dfrac{{{\text{Distance between total rungs}}}}{{{\text{Distance between two rungs}}}} + 1$, (plus one because of the bottom rung and thereafter all rungs are 25 cm apart).

$\therefore $Total number of rungs${\text{ = }}\dfrac{{250}}{{25}} + 1 = 11$

Now as the length of the rungs decrease uniformly therefore they will form an A.P.

So the A.P becomes $\left( {45,.................,25} \right)$

So, first term $\left( {{a_1}} \right)$of an A.P${\text{ = 45}}$, last term $\left( {{a_n}} \right)$of an A.P${\text{ = 25}}$, and number of terms in this A.P$ = 11$

Now, as we know last term of this series is written as

${a_n} = {a_1} + \left( {n - 1} \right)d$, Where d is the common difference.

$ \Rightarrow d = \dfrac{{{a_n} - {a_1}}}{n} + 1 = \dfrac{{25 - 45}}{{11}} + 1 = \dfrac{{ - 20}}{{11}} + 1 = \dfrac{{ - 9}}{{11}}$

So, the length of the rungs decrease uniformly by $\dfrac{{ - 9}}{{11}}cm$

The length of the wood required for the rungs equals the sum of all the terms of this A.P

${S_n} = 45 + \left( {45 - \dfrac{9}{{11}}} \right) + \left( {45 - \dfrac{9}{{11}} - \dfrac{9}{{11}}} \right) + .............. + 25$

Therefore sum of this A.P${\text{ = }}{{\text{S}}_n} = \dfrac{n}{2}\left( {{a_1} + {a_l}} \right)$

$ \Rightarrow {{\text{S}}_n} = \dfrac{{11}}{2}\left( {45 + 25} \right) = 11 \times 35 = 385cm$

Therefore the length of the wood required for the rungs${\text{ = 385cm}}$.

Note: -In such types of questions first find out the total numbers of rungs, then the key concept is that the length of the rungs decrease uniformly so, they will form an A.P so, the length of the wood required for the rungs equals the sum of all the terms of this A.P, so apply the formula of sum of an A.P which is stated above, we will get the required answer.

Recently Updated Pages

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

What happens when entropy reaches maximum class 11 chemistry JEE_Main

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Draw a diagram of nephron and explain its structur class 11 biology CBSE

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Write the difference between soap and detergent class 10 chemistry CBSE

Give 10 examples of unisexual and bisexual flowers

Differentiate between calcination and roasting class 11 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is the difference between anaerobic aerobic respiration class 10 biology CBSE

a Why did Mendel choose pea plants for his experiments class 10 biology CBSE