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To find the 5th term in the geometric sequence 0125, 025, 05, we first need to identify the pattern in the sequence. The given terms can be expressed as numbers: 125, 25, and 5. A geometric sequence is defined by the property that each term is obtained by multiplying the previous term by a constant, … Read more

To define a function using the variables a, b, c, x, and y with the constraint that c lies between 0 and 1, we need to establish a relationship that incorporates these variables appropriately. Let’s consider a simple mathematical function: f(a, b, c, x, y) = a * x + b * y * c … Read more

To find the equation of the line that passes through the points (2, 2) and (1, 4), we first need to determine the slope of the line. The formula for the slope (m) between two points (x1, y1) and (x2, y2) is given by: m = (y2 – y1) / (x2 – x1) Substituting our … Read more

To rewrite the function g(x) = 4x² + 88x in vertex form, we will complete the square. The vertex form of a quadratic function is generally written as: g(x) = a(x – h)² + k where (h, k) is the vertex of the parabola represented by the function. Let’s go through the steps: Start with … Read more

To determine which triangle is similar to triangle ABC based on the provided trigonometric ratios, we first need to analyze the given values of sine, cosine, and tangent. We know the relationships for a triangle’s angles: Sine: sin(A) = opposite / hypotenuse Cosine: cos(A) = adjacent / hypotenuse Tangent: tan(A) = opposite / adjacent Given: … Read more

To determine the probability of a uniformly distributed random variable, we use the properties of uniform distributions. In this case, the random variable x is uniformly distributed between 70 and 90. This means that every value within this interval has an equal likelihood of occurring. The total length of the interval where x can take … Read more

Understanding Asymptotes of the Function y = tan(3x^4) An asymptote is a line that a graph approaches as it heads towards infinity. For the function y = tan(3x^4), we will primarily focus on vertical asymptotes, which occur at points where the function approaches infinity. Identifying Vertical Asymptotes The tangent function, tan(x), has vertical asymptotes at … Read more

To find the complex zeros of the polynomial function f(x) = x3 + x2 – 6x – 6, we can start by applying the Rational Root Theorem and synthetic division to identify any potential rational roots. After evaluating several possible rational roots, we find that x = 2 is a root. We can perform synthetic … Read more

To set up the given statement as a system of equations, we first need to translate the expressions into mathematical equations. The original expression states: one-fifth of x plus two is equal to one-third of x plus eight We can express this in mathematical terms: One-fifth of x can be written as: \( \frac{1}{5} x … Read more

A leap year consists of 366 days, which is one day more than a typical year that has 365 days. The additional day is added to the month of February, making it 29 days long instead of the usual 28 days. The concept of a leap year is rooted in the need to keep our … Read more