3 32. Find The Area Of The Rectangular Park That Is 36 M Long And 16 M Broad. 33. Express Each Of The Following In The Powers Of Their Prime Factors. A) 1250 B) 2048 C) 1080 34. By What Number Should (3)-¹ Be Multiplied So That The Product Is (15)-¹?
Problem 1: Finding the Area of a Rectangular Park
Introduction
In this problem, we will find the area of a rectangular park that is 36 m long and 16 m broad. The area of a rectangle is given by the formula: Area = Length × Breadth.
Solution
To find the area of the rectangular park, we will multiply its length and breadth.
# Define the length and breadth of the rectangular park
length = 36 # in meters
breadth = 16 # in meters
area = length * breadth
print("The area of the rectangular park is:", area, "square meters")
When we run this code, we get the following output:
The area of the rectangular park is: 576 square meters
Therefore, the area of the rectangular park is 576 square meters.
Problem 2: Expressing Numbers in Powers of Prime Factors
Introduction
In this problem, we will express each of the following numbers in powers of their prime factors:
a) 1250 b) 2048 c) 1080
Solution
To express a number in powers of its prime factors, we will find the prime factorization of the number.
a) 1250
The prime factorization of 1250 is:
1250 = 2 × 625 = 2 × 5^4 = 2 × 5^4
Therefore, 1250 can be expressed as 2 × 5^4.
b) 2048
The prime factorization of 2048 is:
2048 = 2^11
Therefore, 2048 can be expressed as 2^11.
c) 1080
The prime factorization of 1080 is:
1080 = 2^3 × 135 = 2^3 × 3^3 × 5 = 2^3 × 3^3 × 5
Therefore, 1080 can be expressed as 2^3 × 3^3 × 5.
Problem 3: Finding the Number to Multiply (3)-¹
Introduction
In this problem, we will find the number to multiply (3)-¹ so that the product is (15)-¹.
Solution
To find the number to multiply (3)-¹, we will use the following formula:
(3)-¹ × x = (15)-¹
We can rewrite (15)-¹ as:
(15)-¹ = (3 × 5)-¹ = 5-¹ × 3-¹
Now, we can equate the two expressions:
(3)-¹ × x = 5-¹ × 3-¹
To find x, we can divide both sides by (3)-¹:
x = 5-¹ × 3-¹ ÷ (3)-¹ = 5-¹
Therefore, the number to multiply (3)-¹ is 5-¹.
Conclusion
In this article, we have solved three mathematical problems:
- Finding the area of a rectangular park that is 36 m long and 16 m broad.
- Expressing each of the following numbers in powers of their prime factors: 1250, 2048, and 1080.
- Finding the number to multiply (3)-¹ so that the product is (15)-¹.
Problem 1: Finding the Area of a Rectangular Park
Q: What is the formula to find the area of a rectangle?
A: The formula to find the area of a rectangle is: Area = Length × Breadth.
Q: How do I calculate the area of a rectangular park that is 36 m long and 16 m broad?
A: To calculate the area of the rectangular park, you can multiply its length and breadth. The Python code to do this is:
# Define the length and breadth of the rectangular park
length = 36 # in meters
breadth = 16 # in meters
area = length * breadth
print("The area of the rectangular park is:", area, "square meters")
Q: What is the area of the rectangular park?
A: The area of the rectangular park is 576 square meters.
Problem 2: Expressing Numbers in Powers of Prime Factors
Q: What is the prime factorization of 1250?
A: The prime factorization of 1250 is: 2 × 5^4.
Q: What is the prime factorization of 2048?
A: The prime factorization of 2048 is: 2^11.
Q: What is the prime factorization of 1080?
A: The prime factorization of 1080 is: 2^3 × 3^3 × 5.
Q: How do I express a number in powers of its prime factors?
A: To express a number in powers of its prime factors, you need to find the prime factorization of the number.
Problem 3: Finding the Number to Multiply (3)-¹
Q: What is the formula to find the number to multiply (3)-¹ so that the product is (15)-¹?
A: The formula to find the number to multiply (3)-¹ is: (3)-¹ × x = (15)-¹.
Q: How do I find the number to multiply (3)-¹?
A: To find the number to multiply (3)-¹, you can use the formula: (3)-¹ × x = (15)-¹. You can rewrite (15)-¹ as: (15)-¹ = (3 × 5)-¹ = 5-¹ × 3-¹. Then, you can equate the two expressions and solve for x.
Q: What is the number to multiply (3)-¹?
A: The number to multiply (3)-¹ is 5-¹.
Frequently Asked Questions (FAQs)
Q: What is the difference between a prime number and a composite number?
A: A prime number is a positive integer that is divisible only by itself and 1. A composite number is a positive integer that is divisible by at least one other positive integer.
Q: How do I find the prime factorization of a number?
A: To find the prime factorization of a number, you need to divide the number by the smallest prime number, which is 2, and continue dividing until you cannot divide the number by 2 anymore. Then, you can divide the number by the next smallest prime number, which is 3, and continue dividing until you cannot divide the number by 3 anymore. You can repeat this process until you have found all the prime factors of the number.
Q: What is the formula to find the area of a triangle?
A: The formula to find the area of a triangle is: Area = (base × height) / 2.
Conclusion
In this article, we have answered some frequently asked questions about mathematical problems and solutions. We have covered topics such as finding the area of a rectangle, expressing numbers in powers of prime factors, and finding the number to multiply (3)-¹. We have also provided Python code to solve some of the problems.