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When a balloon rises vertically above a level, straight road at a constant rate of 1 foot per second, several interesting dynamics are at play. Firstly, the balloon’s ascent is steady, meaning it maintains its speed without any sudden accelerations or decelerations. This steady motion can be described mathematically as a linear function of time. … Read more

Understanding Similar Triangles When we say that two triangles are similar, it means that their corresponding angles are equal, and the lengths of their corresponding sides are proportional. Let’s break this down step-by-step to find the value of x when we have two similar triangles. Step 1: Identify the Corresponding Parts Assuming we have two … Read more

To solve the equation sin(2x) * sin(x) = 0, we need to find the values of x that make either sin(2x) or sin(x) equal to zero. ### Step 1: Set Each Factor to Zero 1. sin(2x) = 0 The sine function equals zero at integer multiples of π: So, we set 2x = nπ, where … Read more

An irrational number is a type of real number that cannot be expressed as a fraction where both the numerator and the denominator are integers. In simpler terms, if you can’t write a number in the form of a/b where a and b are whole numbers (with b not being zero), then it’s irrational. To … Read more

To find the height of an equilateral triangle, we can use the formula: Height (h) = (√3 / 2) * side length In this case, the side length of the triangle is given as 6 cm. Plugging this value into the formula: Height (h) = (√3 / 2) * 6 cm This simplifies to: Height … Read more

To determine the factors of 729 when expressed as x3, we first need to understand what 729 represents in terms of its prime factorization. 729 can be expressed mathematically as: 729 = 9 x 81 = 9 x 9 x 9 = 93 To break it down further: 9 can be represented as 32. So … Read more

Solving for x in the equation 3/4 * x = 5/8 + 4x To solve the equation 3/4 * x = 5/8 + 4x, we’ll go through the steps carefully: Isolate the variable x: First, we want to get all terms involving x on one side of the equation. To do this, we can subtract … Read more

The sequence you’ve provided is 4, 12, 6, 9, 8, 6. To find the next number in this series, let’s first analyze the pattern: The first number is 4. The second number, 12, is 4 multiplied by 3 (4 x 3 = 12). The third number, 6, can be seen as 12 divided by 2 … Read more

To determine which equations have both x = 5 and x = -5 as solutions, we need to consider the characteristics of the equations that yield these values. A polynomial equation can have multiple roots, and if we want both x = 5 and x = -5 to be solutions, we can formulate an equation … Read more