Problem

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

Key Equations

Solution

First, add 7 to both sides of the equation: 4x - 7 + 7 = 21 + 7, which simplifies to 4x = 28. Next, divide both sides by 4: 4x ÷ 4 = 28 ÷ 4, resulting in x = 7.

Explanation

This method works because we use inverse operations to isolate the variable. By adding and then dividing, we maintain the equality of the equation while simplifying it to find the value of 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

Start by substituting the values into the formula: V = π(3 cm)²(5 cm). Calculate the area of the base: (3 cm)² = 9 cm². Then multiply by the height: V = π × 9 cm² × 5 cm = 45π cm³. 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 effectively captures the three-dimensional space occupied by the cylinder.

Problem

Determine the length of the hypotenuse of a right triangle with legs measuring 6 cm and 8 cm. Use the Pythagorean theorem: a² + b² = c².

Key Equations

Solution

Let a = 6 cm and b = 8 cm. According to the theorem, we calculate: 6² + 8² = c². This gives us 36 + 64 = c², simplifying to 100 = c². To find c, take the square root: c = √100 = 10 cm.

Explanation

The Pythagorean theorem relates the lengths of the sides of a right triangle. It allows us to calculate the hypotenuse when the lengths of the other two sides are known, demonstrating the relationship between these dimensions.

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 an if-else conditional statement. It iterates through the numbers 1 to 10, checking each number to determine if it is even or odd. The modulo operator (%) is used to check the remainder when dividing by 2, which helps classify the number. This is a fundamental programming concept that helps in decision-making within loops.

Use Case

This code can be useful in applications where categorizing data is necessary, such as sorting numbers in a game or filtering user input based on specific criteria.

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 calculate_area(base, height):  # Function to calculate area of a triangle
    return 0.5 * base * height  # Area formula

# Example usage
base = 10  # Base of the triangle
height = 6  # Height of the triangle
area = calculate_area(base, height)  # Call the function
print(f'The area of the triangle is {area} cm²')  # Output the result

Explanation

This code defines a function named 'calculate_area' that takes two parameters: base and height. It uses the formula for the area of a triangle (Area = 1/2 * base * height) to compute the area and returns the result. Functions are essential for code reusability and organization, allowing for cleaner and more maintainable code.

Use Case

This function can be applied in various fields such as architecture, engineering, and education, where calculating the area of triangular shapes is necessary for design and analysis.

Output

The area of the triangle is 30.0 cm²

Code

def double_numbers(numbers):  # Function to double each number in a list
    doubled = []  # Initialize an empty list for results
    for number in numbers:  # Iterate through each number in the input list
        doubled.append(number * 2)  # Double the number and add to results
    return doubled  # Return the list of doubled numbers

# Example usage
input_numbers = [1, 2, 3, 4, 5]  # List of numbers
result = double_numbers(input_numbers)  # Call the function
print(f'Doubled numbers: {result}')  # Output the result

Explanation

This code snippet defines a function 'double_numbers' that takes a list of numbers as input. It iterates through each number, doubles it, and appends the result to a new list. Finally, it returns the list of doubled numbers. This demonstrates the use of lists, iteration, and functions, which are core concepts in programming.

Use Case

This function can be useful in data processing tasks where numerical values need to be scaled or transformed, such as in financial calculations or statistical analysis.

Output

Doubled numbers: [2, 4, 6, 8, 10]