To understand how many square inches are in one square foot, we need to start with some basic measurements. A square foot is defined as a square with each side measuring one foot in length. Now, since there are 12 inches in one foot, we can calculate the area of the square foot in square … Read more
The end behavior of a polynomial function gives us a glimpse into how the function behaves as the input values (x) approach positive or negative infinity. For the function f(x) = x³ – 2x² + 4x – 5, the dominant term is x³, because it has the highest degree. As x approaches positive infinity (x … Read more
To find the general solution for the differential equation dy/dx = y * e^(5x), we will use the method of separation of variables. This method allows us to rearrange the equation so that each variable (y and x) is on its own side. First, we can rearrange the equation: dy/dx = y * e^(5x) We … Read more
To determine if two numbers can both have a difference of 8 and a sum of 1, we can set up a system of equations. Let’s denote the two numbers as x and y. According to the given conditions, we can write the following equations: Equation 1 (sum): x + y = 1 Equation 2 … Read more
To find the point on the plane represented by the equation 2x + y + 2z = 20 that is closest to the origin, we can employ the method of Lagrange multipliers or distance optimization techniques. Let’s go through the steps: 1. **Understanding the problem:** We need to minimize the distance from the origin to … Read more
To find the equation of a parabola given the vertex and the focus, we can start by identifying the key components from the given data. In this case, the vertex is at the point (2, 5) and the focus is at (2, 6). Since the x-coordinates of both the vertex and focus are the same, … Read more
To find the value of the cotangent of the acute angle θ given that sin(θ) = 8/9, we can follow these steps: 1. **Understand the definition of cotangent**: The cotangent of an angle θ is the reciprocal of the tangent. Therefore, we can express cot(θ) as: cot(θ) = cos(θ) / sin(θ) 2. **Use the Pythagorean … Read more
Finding the Volume of the Solid To determine the volume of the solid in the first octant bounded by the cylinder defined by z = 16x^2 and the plane y = 5, we first need to visualize the region we are working with. The solid is present in the first octant, where x, y, and … Read more
The vertex of a quadratic function is an important point that represents the highest or lowest value of the function, depending on the direction of the parabola. To find the vertex of the quadratic equation in the standard form, we can use the equation: y = ax^2 + bx + c. In your case, the … Read more
Finding the Possible Rational Zeros To determine the possible rational zeros of the polynomial function f(x) = x^4 + 6x^3 + 3x^2 + 17x – 15, we can use the Rational Root Theorem. This theorem states that if a polynomial has a rational solution (p/q), then: p (the numerator) is a factor of the constant … Read more