Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Explore a wealth of knowledge from professionals across various disciplines on our comprehensive Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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].
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
