The sum of 5 consecutive integers is 270. What is the second number in this sequence?
To find the second number in a sequence of 5 consecutive integers that sum up to 270, we can denote the first integer as x. The next four consecutive integers will then be x + 1, x + 2, x + 3, and x + 4. The sum of these integers can be expressed as: […]
What are the coordinates of point (x, y) corresponding to angle 8 on the unit circle if the measure of angle 8 is 60 degrees?
To find the coordinates of point (x, y) on the unit circle corresponding to a given angle, we use the cosine and sine functions. The unit circle has a radius of 1, so the coordinates can be calculated as: x = cos(θ) and y = sin(θ) Given that the measure of angle 8 is 60 […]
A circle is centered at (3, 8) and is tangent to the x-axis. What is its equation in standard form?
To find the equation of the circle, we start by identifying two key elements based on the information given: the center of the circle and the radius. The center of the circle is given as (3, 8). The radius can be determined because the circle is tangent to the x-axis. The distance from the center […]
Find the Maclaurin Series Expansion of f(x) = cos(3x)
The Maclaurin series is a special case of the Taylor series, which is used to expand functions around the point x=0. To find the Maclaurin series expansion of the function f(x) = cos(3x), we will use the formula for the Taylor series: f(x) = f(0) + f'(0)x + rac{f”(0)}{2!}x^2 + rac{f”'(0)}{3!}x^3 + … + rac{f^{(n)}(0)}{n!}x^n […]
What is the derivative of sin(1/x)?
To find the derivative of the function sin(1/x), we will use the chain rule of differentiation. Let y = sin(1/x). The chain rule states that if you have a composite function f(g(x)), the derivative is f'(g(x)) imes g'(x). In this case, we have: f(u) = sin(u) where u = 1/x So, f'(u) = cos(u) And, […]
Find the Angle Between the Given Vectors to the Nearest Tenth of a Degree: u (2, 4) and v (3, 8)
To find the angle θ between the two vectors u and v, we can use the formula: cos(θ) = (u • v) / (|u| |v|) First, we need to calculate the dot product of the vectors u and v: u • v = (2 * 3) + (4 * 8) = 6 + 32 = […]
Which is not an equation of the line that passes through the points (1, 1) and (5, 5)?
The line passing through the points (1, 1) and (5, 5) can be determined by finding its slope and using the point-slope form of a line. First, we calculate the slope (m) of the line: m = (y2 – y1) / (x2 – x1) = (5 – 1) / (5 – 1) = 4 / […]
Solve the equation t² – t – 12 by factoring
To solve the equation t² – t – 12 = 0 by factoring, we first need to factor the quadratic expression on the left side. We are looking for two numbers that multiply to -12 (the constant term) and add up to -1 (the coefficient of the linear term). After examining the factors of -12, […]
How to Find Functions f(x) and g(x) Given y = f(g(x)) with y = 102x + 9?
To find the functions f(x) and g(x) such that the function can be described as y = f(g(x)), given y = 102x + 9, we can take the following approach: First, let’s define g(x) in a simple linear form. A common choice is to let g(x) = ax + b, where a and b are […]
Let A be the Set of All Points in a Plane and O be the Origin. Show That the Relation to P Q P Q A and OP OQ is an Equivalence Relation.
To show that the relation defined on the set A of all points in a plane (where O is the origin) is an equivalence relation, we need to verify that it satisfies three properties: reflexivity, symmetry, and transitivity. 1. Reflexivity: A relation is reflexive if every element is related to itself. For a point P […]