Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Get quick and reliable solutions to your questions from a community of experienced experts on our platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To solve the problem, we need to identify the age of Mr. X last year that satisfies two conditions: it must be the square of a number last year and the cube of a number next year. Then, we will determine the minimum number of years Mr. X must wait for his age to become the cube of a number again.
### Step-by-Step Solution:
1. Identify Mr. X's Age Last Year and This Year:
- We need to find the smallest age [tex]\( n \)[/tex] such that:
- [tex]\( n^2 \)[/tex] is Mr. X's age last year.
- [tex]\( n^2 + 2 \)[/tex] is the cube of a number.
2. Mathematical Representation:
- Let [tex]\( n \)[/tex] be an integer. Then:
- Mr. X's age last year = [tex]\( n^2 \)[/tex]
- Mr. X's age this year = [tex]\( n^2 + 1 \)[/tex]
- Mr. X's age next year = [tex]\( n^2 + 2 \)[/tex]
3. Condition for Next Year’s Age:
- Mr. X’s age next year must be a perfect cube.
- We need to find the smallest [tex]\( n \)[/tex] such that [tex]\( (n^2 + 2) \)[/tex] is a perfect cube.
4. Finding the Valid Age:
- By examining small values of [tex]\( n \)[/tex]:
- For [tex]\( n = 1 \)[/tex]:
- [tex]\( n^2 + 2 = 1 + 2 = 3 \)[/tex] (not a cube)
- For [tex]\( n = 2 \)[/tex]:
- [tex]\( n^2 + 2 = 4 + 2 = 6 \)[/tex] (not a cube)
- Continue this process until:
- For [tex]\( n = 5 \)[/tex]:
- [tex]\( n^2 + 2 = 25 + 2 = 27 \)[/tex]
- [tex]\( 27 \)[/tex] is indeed [tex]\( 3^3 \)[/tex], a perfect cube.
- Therefore, Mr. X's age last year was [tex]\( 5^2 = 25 \)[/tex] years old.
- Mr. X's age next year is [tex]\( 27 \)[/tex], a perfect cube.
5. Finding Number of Years until Mr. X's Age is a Cube Again:
- We start from his current age (which is 26 years) and increment year by year, checking when it will next be a cube.
- From the solution, we found the next instance where Mr. X's age is a cube occurs 37 years after his current age.
So, the least number of years Mr. X must wait to become the cube of a number again is:
Answer: (B) 38
It results in the next occurrence of a perfect cube age after 27 being precisely 27 years added to 10 giving - Corrected final answer is: 27 + 37. Thus answer finalized:B. 38 years.
### Step-by-Step Solution:
1. Identify Mr. X's Age Last Year and This Year:
- We need to find the smallest age [tex]\( n \)[/tex] such that:
- [tex]\( n^2 \)[/tex] is Mr. X's age last year.
- [tex]\( n^2 + 2 \)[/tex] is the cube of a number.
2. Mathematical Representation:
- Let [tex]\( n \)[/tex] be an integer. Then:
- Mr. X's age last year = [tex]\( n^2 \)[/tex]
- Mr. X's age this year = [tex]\( n^2 + 1 \)[/tex]
- Mr. X's age next year = [tex]\( n^2 + 2 \)[/tex]
3. Condition for Next Year’s Age:
- Mr. X’s age next year must be a perfect cube.
- We need to find the smallest [tex]\( n \)[/tex] such that [tex]\( (n^2 + 2) \)[/tex] is a perfect cube.
4. Finding the Valid Age:
- By examining small values of [tex]\( n \)[/tex]:
- For [tex]\( n = 1 \)[/tex]:
- [tex]\( n^2 + 2 = 1 + 2 = 3 \)[/tex] (not a cube)
- For [tex]\( n = 2 \)[/tex]:
- [tex]\( n^2 + 2 = 4 + 2 = 6 \)[/tex] (not a cube)
- Continue this process until:
- For [tex]\( n = 5 \)[/tex]:
- [tex]\( n^2 + 2 = 25 + 2 = 27 \)[/tex]
- [tex]\( 27 \)[/tex] is indeed [tex]\( 3^3 \)[/tex], a perfect cube.
- Therefore, Mr. X's age last year was [tex]\( 5^2 = 25 \)[/tex] years old.
- Mr. X's age next year is [tex]\( 27 \)[/tex], a perfect cube.
5. Finding Number of Years until Mr. X's Age is a Cube Again:
- We start from his current age (which is 26 years) and increment year by year, checking when it will next be a cube.
- From the solution, we found the next instance where Mr. X's age is a cube occurs 37 years after his current age.
So, the least number of years Mr. X must wait to become the cube of a number again is:
Answer: (B) 38
It results in the next occurrence of a perfect cube age after 27 being precisely 27 years added to 10 giving - Corrected final answer is: 27 + 37. Thus answer finalized:B. 38 years.
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
