Problem

Solve the equation 4x - 7 = 21. This equation requires isolating the variable x to find its value.

Key Equations

Solution

To solve for x, first add 7 to both sides: 4x = 21 + 7. This simplifies to 4x = 28. Next, divide both sides by 4: x = 28 ÷ 4. Thus, x = 7.

Explanation

This method works because we are using inverse operations to isolate the variable. By performing the same operation on both sides of the equation, we maintain equality and can solve for x.

Problem

Find the volume of a cylinder with a radius of 3 cm and a height of 5 cm. Use the formula V = πr²h.

Key Equations

Solution

First, calculate the area of the base: A = π(3)² = π × 9 = 9π. Then, multiply by the height: V = 9π × 5 = 45π. Therefore, the volume is approximately 141.37 cm³ when using π ≈ 3.14.

Explanation

The volume of a cylinder is derived from the area of its circular base multiplied by its height. This formula allows us to determine how much space is contained within the cylinder.

Problem

Use the Pythagorean theorem to find the length of the hypotenuse of a right triangle with legs measuring 6 cm and 8 cm.

Key Equations

Solution

According to the Pythagorean theorem, a² + b² = c², where c is the hypotenuse. Here, a = 6 cm and b = 8 cm. So, we calculate: 6² + 8² = c², which is 36 + 64 = c². Thus, c² = 100, leading to c = √100 = 10 cm.

Explanation

The Pythagorean theorem is fundamental in geometry, relating the lengths of the sides of a right triangle. It allows us to find the length of one side when the other two are known, demonstrating the relationship between them.

Code

for i in range(1, 11):  # Loop through numbers 1 to 10
    if i % 2 == 0:  # Check if the number is even
        print(f'{i} is even')  # Print if even
    else:
        print(f'{i} is odd')  # Print if odd

Explanation

This code demonstrates the use of a for loop combined with conditional statements. It iterates through numbers 1 to 10, checking each number to determine if it is even or odd using the modulus operator. The output is formatted to indicate whether each number is even or odd, showcasing basic control flow in Python.

Use Case

This code can be useful in applications where categorizing data into different groups is necessary, such as sorting numbers for statistical analysis or generating reports based on conditions.

Output

1 is odd
2 is even
3 is odd
4 is even
5 is odd
6 is even
7 is odd
8 is even
9 is odd
10 is even

Code

def factorial(n):  # Define a function to calculate factorial
    if n == 0 or n == 1:  # Base case for recursion
        return 1
    else:
        return n * factorial(n - 1)  # Recursive call

# Usage
result = factorial(5)  # Calculate factorial of 5
print(f'The factorial of 5 is {result}')

Explanation

This code defines a recursive function to calculate the factorial of a given number. The base case checks if the number is 0 or 1, returning 1 in those cases. For other numbers, the function calls itself with the number decremented by one, multiplying the results. This illustrates recursion, a fundamental programming concept.

Use Case

Calculating factorials is common in combinatorics, probability, and statistics, making this function useful in mathematical computations and algorithm design.

Output

The factorial of 5 is 120

Code

def process_data(data):  # Function to process a list of numbers
    processed = []  # Initialize an empty list for processed data
    for number in data:  # Iterate over each number in the input list
        processed.append(number ** 2)  # Append the square of each number
    return processed  # Return the processed list

# Usage
numbers = [1, 2, 3, 4, 5]  # Sample list of numbers
squared_numbers = process_data(numbers)  # Process the list
print(f'Squared numbers: {squared_numbers}')

Explanation

This code defines a function that takes a list of numbers, squares each number, and returns a new list containing the squared values. It demonstrates list manipulation and function creation in Python, which are essential for data processing tasks.

Use Case

This function can be applied in data analysis scenarios where transformations of numerical data are required, such as preparing datasets for machine learning models.

Output

Squared numbers: [1, 4, 9, 16, 25]