Answered

Join the growing community of curious minds on IDNLearn.com and get the answers you need. Ask any question and receive comprehensive, well-informed responses from our dedicated team of experts.

Three measuring rods are 40 cm, 50 cm and 80 cm long. Find the shortes
length of cloth which can be measured exactly by any one of these rods.
please

​

Sagot :

Jacob193idn

Answer:

[tex]400\; {\rm cm}[/tex].

Step-by-step explanation:

A measuring rod exactly measures a given length if and only if that length is a multiple of the length of the measuring rod.

For example:

  • A [tex]40\; {\rm cm}[/tex] measuring rod can exactly measure lengths of [tex]40\; {\rm cm}[/tex], [tex]2 \times 40\; {\rm cm} = 80\; {\rm cm}[/tex], etc.
  • A [tex]50\; {\rm cm}[/tex] measuring rod can exactly measure lengths of [tex]50\; {\rm cm}[/tex], [tex]2 \times 50\; {\rm cm} = 100\; {\rm cm}[/tex], etc.
  • A [tex]80\; {\rm cm}[/tex] measuring rod can exactly measure lengths of [tex]80\; {\rm cm}[/tex], [tex]2 \times 80\; {\rm cm} = 160\; {\rm cm}[/tex], etc.

If a certain length needs to be exactly measurable using any one of the three measuring rods, that length needs to be a common multiple of all three measuring rod lengths: [tex]40\; {\rm cm}[/tex], [tex]50\; {\rm cm}[/tex], and [tex]80\; {\rm cm}[/tex]. Since the question is asking for the least possible length satisfying these requirements, the goal is equivalent to finding the least common multiple of these three numbers: [tex]40[/tex], [tex]50[/tex], and [tex]80[/tex].

To find the least common multiple of [tex]40[/tex], [tex]50[/tex], and [tex]80[/tex], start by factoring each of the three numbers:

  • [tex]40 = 2 \times 2 \times 2 \times 5 = 2^{3} \times 5[/tex].
  • [tex]50 = 2 \times 5 \times 5 = 2 \times 5^{2}[/tex].
  • [tex]80 = 2 \times 2 \times 2 \times 2 \times 5 = 2^{4} \times 5[/tex].

The prime factors of these three numbers are [tex]2[/tex] and [tex]5[/tex]. Among these three numbers, the maximum power of [tex]2[/tex] is [tex]4[/tex] while the maximum power of [tex]5[/tex] is [tex]2[/tex]. Hence, the least common multiple of these three numbers would be:

[tex]2^{4} \times 5^{2} = 400[/tex].

Therefore, [tex]400\; {\rm cm}[/tex] would be the shortest possible length that can be measured exactly using any one of the three measuring rods.

  • [tex](40 \; {\rm cm}) \times 10[/tex].
  • [tex](50 \; {\rm cm}) \times 8[/tex].
  • [tex](80 \; {\rm cm}) \times 5[/tex].
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Trust IDNLearn.com for all your queries. We appreciate your visit and hope to assist you again soon.