Find the center and radius of the sphere given by the equation x² + y² + z² – 4x – 2z – 8 = 0
To find the center and radius of the sphere from the given equation, we can rewrite the equation in standard form. The given equation is: x² + y² + z² – 4x – 2z – 8 = 0 We can rearrange this equation to isolate the variables: x² – 4x + y² + z² – […]
Determine if the graph is symmetric about the x axis, the y axis, or the origin for r = 8 cos(3θ)
To determine the symmetry of the polar graph given by the equation r = 8 cos(3θ), we need to analyze it based on the characteristics of polar coordinates. 1. Symmetry about the x-axis: A polar graph is symmetric about the x-axis if replacing θ with -θ results in the same equation. Let’s check this: Substituting […]
If ux = 2x + 3 and vx = 1x, what is the range of uvx?
To find the range of uvx where u(x) = 2x + 3 and v(x) = 1x, we first need to find the expression for uvx. By multiplying these two functions together, we have: uv(x) = u(x) imes v(x) = (2x + 3) imes (1x) = 2x^2 + 3x Now, we want to determine the range […]
If xy = 144 and x + y = 30, what is the value of x and y?
To find the values of x and y, we can start with the two equations we have: xy = 144 x + y = 30 From the second equation, we can express y in terms of x: y = 30 – x Now, we can substitute this expression for y into the first equation: x(30 […]
How to Find the Area of a Rhombus with One Diagonal and Perimeter
To find the area of a rhombus when you have one diagonal and the perimeter, you can follow these steps: 1. **Understand the properties of a rhombus**: A rhombus is a type of quadrilateral where all sides are of equal length. The diagonals of a rhombus bisect each other at right angles. 2. **Determine the […]
What is the probability that a five card poker hand contains cards of five different kinds?
To find the probability that a five card poker hand contains cards of five different kinds, we must first understand what is meant by ‘five different kinds’. This means that we want one card from each of the five different ranks available in a standard deck of cards (which has 13 ranks: Ace, 2, 3, […]
Find the differential dy and evaluate dy for the given values of x and dx: y = 1/x, x = 0, dx = 0.02
To find the differential dy, we start with the function: y = 1/x The differential dy can be found using the formula: dy = y’ * dx First, we need to calculate the derivative y’: y’ = -1/x^2 Now, we plug in the given value of x
How do I solve the system of equations 2x + 3y = 40 and 2x + 2y = 20?
To solve the system of equations, we have: Equation 1: 2x + 3y = 40 Equation 2: 2x + 2y = 20 We can use either substitution or elimination method. Here, we’ll use the elimination method for simplicity. First, we notice that both equations have the term 2x
If the area of the triangle BGC is 28 square units and the centroid of the triangle is G, then find the area of the triangle ABC.
To find the area of triangle ABC, we can use the property of the centroid in relation to the areas of triangles formed with it. The centroid (G) of a triangle divides it into three smaller triangles (BGC, CAG, and ABG), and the area of each of these smaller triangles is equal to one-third of […]
What is the center of a circle whose equation is x² + y² – 12x – 2y + 12 = 0?
To find the center of a circle from its equation, we need to rewrite the equation in the standard form of a circle, which is: (x – h)² + (y – k)² = r² where (h, k) is the center of the circle and r is the radius. Given the equation: x² + y² – […]