Answer :
To determine the total length of wood required to construct the 50 rungs of the ladder, we use the concept of an arithmetic series. Here's a step-by-step approach:
1. Identify the initial conditions:
- The first rung of the ladder is 1 meter long.
- Each rung is 12.5 mm shorter than the rung beneath it. To work consistently in meters, convert this measurement:
[tex]\(12.5 \text{ mm} = 12.5 / 1000 = 0.0125 \text{ m}\)[/tex].
2. Determine the total number of rungs:
- The ladder has 50 rungs.
3. Understand the sequence of lengths:
- This is an arithmetic sequence, where each subsequent term (rung length) is diminished by 0.0125 m from the previous term.
4. Set up the arithmetic sequence:
- First term ([tex]\( a_1 \)[/tex]) is the length of the first rung: [tex]\( 1 \text{ m} \)[/tex].
- Common difference ([tex]\( d \)[/tex]) is the reduction in length at each step: [tex]\(-0.0125 \text{ m} \)[/tex].
5. Calculate the total length of the rungs using the formula for the sum of an arithmetic series:
[tex]\[
S_n = \frac{n}{2} \times (2a_1 + (n-1) \times d)
\][/tex]
- [tex]\( n \)[/tex] is the number of rungs: [tex]\( 50 \)[/tex].
- Plug in the values:
[tex]\[
S_{50} = \frac{50}{2} \times (2 \times 1 + (50-1) \times (-0.0125))
\][/tex]
6. Calculate the expression:
[tex]\[
S_{50} = 25 \times (2 - 49 \times 0.0125)
\][/tex]
[tex]\[
= 25 \times (2 - 0.6125)
\][/tex]
[tex]\[
= 25 \times 1.3875
\][/tex]
[tex]\[
= 34.6875
\][/tex]
Thus, the total length of wood needed to make the 50 rungs is [tex]\( 34.6875 \text{ meters} \)[/tex].
1. Identify the initial conditions:
- The first rung of the ladder is 1 meter long.
- Each rung is 12.5 mm shorter than the rung beneath it. To work consistently in meters, convert this measurement:
[tex]\(12.5 \text{ mm} = 12.5 / 1000 = 0.0125 \text{ m}\)[/tex].
2. Determine the total number of rungs:
- The ladder has 50 rungs.
3. Understand the sequence of lengths:
- This is an arithmetic sequence, where each subsequent term (rung length) is diminished by 0.0125 m from the previous term.
4. Set up the arithmetic sequence:
- First term ([tex]\( a_1 \)[/tex]) is the length of the first rung: [tex]\( 1 \text{ m} \)[/tex].
- Common difference ([tex]\( d \)[/tex]) is the reduction in length at each step: [tex]\(-0.0125 \text{ m} \)[/tex].
5. Calculate the total length of the rungs using the formula for the sum of an arithmetic series:
[tex]\[
S_n = \frac{n}{2} \times (2a_1 + (n-1) \times d)
\][/tex]
- [tex]\( n \)[/tex] is the number of rungs: [tex]\( 50 \)[/tex].
- Plug in the values:
[tex]\[
S_{50} = \frac{50}{2} \times (2 \times 1 + (50-1) \times (-0.0125))
\][/tex]
6. Calculate the expression:
[tex]\[
S_{50} = 25 \times (2 - 49 \times 0.0125)
\][/tex]
[tex]\[
= 25 \times (2 - 0.6125)
\][/tex]
[tex]\[
= 25 \times 1.3875
\][/tex]
[tex]\[
= 34.6875
\][/tex]
Thus, the total length of wood needed to make the 50 rungs is [tex]\( 34.6875 \text{ meters} \)[/tex].