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When we say that two ratios are equivalent, we mean that they express the same relationship between two quantities, even if the numbers used in the ratios are different. In mathematical terms, two ratios a:b and c:d are considered equivalent if: The cross products of the ratios are equal, which mathematically can be expressed as: … Read more

The function y = x2 – 25 is a polynomial function. Polynomial functions are differentiable everywhere in their domain, which is all real numbers. This means that for the equation y = x2 – 25, it is differentiable at any value of x. In interval notation, we can express this as: (-∞, ∞) This implies … Read more

To find the derivative of the function g(t) = 9t using the definition of derivative, we start with the formal definition: Definition of Derivative: The derivative of a function g(t) at a point t is defined as: g'(t) = lim (h -> 0) [(g(t + h) – g(t)) / h] Step 1: Substitute the function … Read more

Finding the Surface Area of the Sphere Above the Cone The problem requires calculating the surface area of a region on the sphere where it is above the cone. Let’s break down the steps needed to solve this in detail. 1. Understand the Geometry We have a sphere defined by the equation: x² + y² … Read more

To find the area of a triangle, you can use the formula: Area = (Base × Height) / 2 In this case, the base of the triangle is 10 inches and the height (or altitude) is 16 inches. Plugging these values into the formula, you get: Area = (10 inches × 16 inches) / 2 … Read more

Finding the Area of the Largest Inscribed Rectangle in a Semi-Circle To determine the area of the largest rectangle inscribed in a semi-circle with a radius of 5, we can use some geometry and calculus. 1. Understanding the Geometry Consider a semi-circle with center at the origin (0,0) in the Cartesian plane, extending from -5 … Read more

To solve for the value of p in the inverse variation equation given by p = 8v, we can follow these steps: Identify the value of v: In this case, we have v = 14. Substitute the value of v into the equation: We replace v in the equation with 14. So the equation becomes: … Read more

To simplify the expression x1/2 divided by x2/7, we can use the laws of exponents. Specifically, the rule states that when you divide like bases, you subtract the exponents. Therefore, we can express this as: x1/2 / x2/7 = x1/2 – 2/7 Now, to perform the subtraction of the exponents 1/2 and 2/7, we first … Read more

Finding f(g(x)) To find f(g(x)), we need to substitute g(x) into the function f(x). Let’s break this down step by step: Step 1: Define the functions We have two functions: f(x) = 4x + 7 g(x) = 3×2 Step 2: Substitute g(x) into f(x) Now, we’ll substitute g(x) into f(x): f(g(x)) = f(3×2) Step 3: … Read more

To evaluate this expression involving angles in different quadrants, we first need to understand the trigonometric properties in each quadrant. 1. **Understanding the Quadrants**: In Quadrant III, both sine and cosine values are negative, while in Quadrant II, sine values are positive, and cosine values are negative. 2. **Evaluate Individual Components**: tan(8): The tangent function … Read more