What does it mean for two ratios to be equivalent?

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

How do you find the surface area of the part of the sphere defined by the equation x² + y² + z² = 81 that is located above the cone defined by z = √(x² + y²)?

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

What is the area of the largest rectangle that can be inscribed in a semi-circle with a radius of 5?

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

How do you evaluate the expression tan(8) * f * cos(8) * 13 * 8 in Quadrant III and sin(f) * 14 * f in Quadrant II?

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