Unlocking Square Roots And Areas A Comprehensive Guide To Section B Problems

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This comprehensive guide dives deep into Section B, focusing on the fascinating world of square roots and areas. We'll dissect problems involving finding the side length of a square given its area and master the division method for calculating square roots. This section aims to equip you with the skills and understanding necessary to confidently tackle these mathematical challenges. So, let's embark on this journey of mathematical discovery!

Q1: Unveiling the Side Length of a Square

Understanding the Relationship Between Area and Side Length

At the heart of this problem lies the fundamental relationship between the area of a square and the length of its sides. Remember, a square is a quadrilateral with four equal sides and four right angles. The area of a square is calculated by squaring the length of one of its sides. Mathematically, this can be represented as:

Area = side * side = side^2

Conversely, if we know the area of a square, we can find the length of one side by taking the square root of the area. This is the core concept we'll use to solve this problem.

Side = √Area

Applying the Concept to the Problem

Our problem states that the area of the square is 680625 cm². To find the length of one side, we need to calculate the square root of this number. This might seem daunting at first, especially with such a large number, but we can use different methods to find the square root, including prime factorization or the long division method. For larger numbers, the long division method is generally more efficient. Let's delve into how we can apply it to this specific problem.

Calculating the Square Root Using Prime Factorization

Prime factorization involves breaking down a number into its prime factors, which are prime numbers that multiply together to give the original number. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself (e.g., 2, 3, 5, 7, 11, etc.). To find the square root using prime factorization, we first find the prime factors of the area, and then we group these factors into pairs. For each pair of identical prime factors, we take one factor out of the square root. If any factors are left unpaired, the number is not a perfect square.

Let's find the prime factorization of 680625:

680625 = 3 * 3 * 5 * 5 * 5 * 5 * 17 * 17

Now, let’s group the identical prime factors into pairs:

680625 = (3 * 3) * (5 * 5) * (5 * 5) * (17 * 17)

Taking one factor from each pair, we get:

√680625 = 3 * 5 * 5 * 17 = 1275

Therefore, the length of one side of the square is 1275 cm.

Solution

The length of one side of the square is 1275 cm.

Q2: Mastering the Division Method for Square Roots

Understanding the Division Method

The division method, also known as the long division method for square roots, is a powerful technique for finding the square root of a number, especially when dealing with large numbers or numbers that are not perfect squares. It's a systematic approach that breaks down the process into manageable steps. The method is based on the algebraic identity (a + b)² = a² + 2ab + b², which helps us approximate the square root iteratively.

(i) Finding the Square Root of 49284

  1. Pair the Digits: Start by pairing the digits of the number from right to left. In this case, 49284 becomes 4 92 84.
  2. Find the Largest Square: Find the largest perfect square less than or equal to the leftmost group (4). The largest perfect square less than or equal to 4 is 4 itself (2² = 4). Write 2 as the divisor and the quotient.
  3. Subtract and Bring Down: Subtract 4 from 4, which leaves 0. Bring down the next pair of digits (92) to form the new dividend 92.
  4. Double the Quotient: Double the current quotient (2) to get 4. This becomes the first part of our new divisor.
  5. Find the Next Digit: We need to find a digit (let's call it 'x') to place next to 4 (making the divisor 4x) such that (4x) * x is less than or equal to 92. By trial and error, we find that 2 is the appropriate digit because (42) * 2 = 84, which is less than 92. Write 2 as the next digit in the quotient and as the digit 'x' in the divisor.
  6. Subtract and Bring Down: Subtract 84 from 92, which leaves 8. Bring down the next pair of digits (84) to form the new dividend 884.
  7. Double the Quotient: Double the current quotient (22) to get 44. This becomes the first part of our new divisor.
  8. Find the Next Digit: We need to find a digit (let's call it 'x') to place next to 44 (making the divisor 44x) such that (44x) * x is less than or equal to 884. By trial and error, we find that 2 is the appropriate digit because (442) * 2 = 884. Write 2 as the next digit in the quotient and as the digit 'x' in the divisor.
  9. Subtract: Subtract 884 from 884, which leaves 0. Since the remainder is 0 and there are no more digits to bring down, the process is complete.

Therefore, the square root of 49284 is 222.

(ii) Finding the Square Root of 46225

  1. Pair the Digits: Pair the digits of the number from right to left. In this case, 46225 becomes 4 62 25.
  2. Find the Largest Square: Find the largest perfect square less than or equal to the leftmost group (4). The largest perfect square less than or equal to 4 is 4 itself (2² = 4). Write 2 as the divisor and the quotient.
  3. Subtract and Bring Down: Subtract 4 from 4, which leaves 0. Bring down the next pair of digits (62) to form the new dividend 62.
  4. Double the Quotient: Double the current quotient (2) to get 4. This becomes the first part of our new divisor.
  5. Find the Next Digit: We need to find a digit (let's call it 'x') to place next to 4 (making the divisor 4x) such that (4x) * x is less than or equal to 62. By trial and error, we find that 1 is the appropriate digit because (41) * 1 = 41, which is less than 62. Write 1 as the next digit in the quotient and as the digit 'x' in the divisor.
  6. Subtract and Bring Down: Subtract 41 from 62, which leaves 21. Bring down the next pair of digits (25) to form the new dividend 2125.
  7. Double the Quotient: Double the current quotient (21) to get 42. This becomes the first part of our new divisor.
  8. Find the Next Digit: We need to find a digit (let's call it 'x') to place next to 42 (making the divisor 42x) such that (42x) * x is less than or equal to 2125. By trial and error, we find that 5 is the appropriate digit because (425) * 5 = 2125. Write 5 as the next digit in the quotient and as the digit 'x' in the divisor.
  9. Subtract: Subtract 2125 from 2125, which leaves 0. Since the remainder is 0 and there are no more digits to bring down, the process is complete.

Therefore, the square root of 46225 is 215.

Solutions

(i) √49284 = 222

(ii) √46225 = 215

Q3: Decoding the Relationship Between Two Numbers and Their Product

Translating the Word Problem into Equations

This problem involves a relationship between two positive numbers and their product. The key to solving it lies in translating the given information into mathematical equations. Let's represent the two positive numbers as 'x' and 'y'. We are given two crucial pieces of information:

  1. The product of the two numbers is 84500: x * y = 84500
  2. One of the numbers is 5 times the other: We can express this as either x = 5y or y = 5x. For the sake of consistency, let's assume x = 5y.

Now we have a system of two equations with two unknowns, which we can solve using various algebraic techniques.

Solving the System of Equations

We have the following system of equations:

  • x * y = 84500
  • x = 5y

We can use the substitution method to solve this system. Substitute the value of x from the second equation into the first equation:

(5y) * y = 84500

This simplifies to:

5y² = 84500

Now, divide both sides by 5:

y² = 16900

To find y, take the square root of both sides:

y = √16900

Calculating the square root, we get:

y = 130

Now that we have the value of y, we can find the value of x using the equation x = 5y:

x = 5 * 130

x = 650

Therefore, the two positive numbers are 130 and 650.

Verification

It's always a good practice to verify our solution. Let's check if our values satisfy the given conditions:

  • Product: 130 * 650 = 84500 (Correct)
  • Relationship: 650 = 5 * 130 (Correct)

Solution

The two positive numbers are 130 and 650.

This detailed breakdown of Section B equips you with the knowledge and techniques to confidently solve problems related to square roots and areas. Remember, practice is key to mastering these concepts. Keep exploring and expanding your mathematical horizons!