What is the greatest common factor of the terms in the polynomial 12x⁴, 27x³, 6x²?
To find the greatest common factor (GCF) of the terms in the polynomial 12x⁴, 27x³, and 6x², we need to identify both the numerical and variable parts of each term. First, let’s break down each coefficient: 12 can be factored into: 2 × 2 × 3 27 can be factored into: 3 × 3 × […]
What is the Radius of a Circle?
The radius of a circle is the distance from the center of the circle to any point on its circumference. It is one of the key measurements in the geometry of circles and plays a crucial role in various circle-related calculations. Mathematically, if you know the diameter of a circle, you can find the radius […]
There are 6 diagonals that can be drawn from one vertex of an octagon true or false?
This statement is false. In an octagon, which has 8 sides, the number of diagonals that can be drawn from one vertex can be calculated using the formula: Diagonals from one vertex = n – 3, where n is the number of sides (or vertices) of the polygon. For an octagon, substituting 8 for n: […]
If x and y vary directly and x is 12 when y is 3, what will be the value of x when y is 9?
To solve for the value of x when y is 9, we first need to understand direct variation. When two variables vary directly, their ratio remains constant. This means we can set up a proportion using the values we know. From the information given, we know that: x = 12 when y = 3 Using […]
How do you solve the system of equations using elimination: 3x + 4y = 9 and 3x + 2y = 9?
To solve this system of equations using the elimination method, we start by writing the two equations clearly: Equation 1: 3x + 4y = 9 Equation 2: 3x + 2y = 9 Next, we notice that both equations have the term 3x. To eliminate x from the equations, we can subtract Equation 2 from Equation […]
How to Find the Linearization L(x) of the Function f(x) at a Point?
To find the linearization, L(x), of the function f(x) at a point a, we follow the formula: L(x) = f(a) + f'(a)(x – a) In this case, we have the function f(x) = x^4 + 5 and we want to find the linearization at the point a = 2. First, we need to evaluate f(a): […]
Find the equation of the line passing through the point (6, 3) that is perpendicular to the line 4x + 5y = 10
To find the equation of the line that is perpendicular to the line given by the equation 4x + 5y = 10 and passes through the point (6, 3), we first need to determine the slope of the original line. We can rewrite the equation in slope-intercept form (y = mx + b). Starting with: […]
What is the variance of a binomial distribution for which n = 100 and p = 0.20?
The variance of a binomial distribution can be calculated using the formula: Variance (σ²) = n * p * (1 – p) In this case, we have: n = 100 (the number of trials) p = 0.20 (the probability of success on each trial) Now, we can substitute the values into the formula: Variance = […]
What is an equation for a sine curve with amplitude 2 and period 4 pi radians?
The equation of a sine curve can be formulated with parameters that define its amplitude and period. In this case, we are looking for a sine function with an amplitude of 2 and a period of 4π radians. The general form of the sine function is: y = A * sin(B(x – C)) + D […]
Let r be the region bounded by the x-axis, the graph of y = x + 1, and the line x = 3. Find the area of the region r.
To find the area of the region r, we need to first understand the boundaries defined in the question. The region is bounded by the x-axis, the line x = 3, and the line represented by the equation y = x + 1. The first step is to identify the points where the line y […]