What is the value of the magnitude of the difference of vectors a and b?
To determine the magnitude of the difference between two vectors, a and b, we first need to find the vector difference, which is given by: a – b Once we have this resulting vector, the magnitude can be calculated using the formula: |a – b| = √((a1 – b1)² + (a2 – b2)² + … […]
Which of the following is the correct graph of the solution to the inequality 13 < 5x + 2 < 28 on a number line with open dots at 3 and 6?
To solve the inequality 13 < 5x + 2 < 28, we start by breaking it into two parts: 5x + 2 > 13 and 5x + 2 < 28. 1. For the first part, 5x + 2 > 13: Subtract 2 from both sides: 5x > 11 Now divide by 5: x > 11/5 […]
Find an equation of the set of all points equidistant from the points A(1, 5, 3) and B(6, 2, 2). Describe the set.
To find the equation of the set of points that are equidistant from the two points A(1, 5, 3) and B(6, 2, 2), we need to determine the set of points P(x, y, z) that satisfy the condition: Distance from A to P = Distance from B to P This can be expressed mathematically as […]
Given the arithmetic sequence an = 4, 3n, 1. What is the domain for n?
To find the domain for n in the arithmetic sequence defined by an = 4, 3n, 1, we first need to clarify the structure of this sequence. This sequence seems to be defined for different values of n. If we interpret this as an explicit formula for a sequence where the nth term depends on […]
What is the y-intercept of the quadratic function f(x) = x^2 – 8x + 3?
The y-intercept of a function is the point where the graph of the function crosses the y-axis. This occurs when the value of x is zero. To find the y-intercept of the quadratic function f(x) = x² – 8x + 3, we need to substitute x with 0 in the function. So, we have: f(0) […]
What is the sum of the arithmetic sequence 8, 15, 22 if there are 26 terms?
To find the sum of the arithmetic sequence, we first need to identify the first term, the common difference, and the number of terms. In this sequence: The first term (t) is 8. The second term is 15, leading to a common difference (d) of 15 – 8 = 7. The number of terms (n) […]
Identify the Open Interval on Which the Function is Increasing or Decreasing: cos(4x) – 3
To determine the interval on which the function f(x) = cos(4x) – 3 is increasing or decreasing, we first need to find its derivative. The derivative of the function is calculated as follows: f'(x) = d/dx[cos(4x) – 3] = -4sin(4x) Now, we set the derivative equal to zero to find the critical points: -4sin(4x) = […]
The measure of an exterior angle of a triangle is equal to the sum of the measure of the two interior opposite angles. Why is this the case?
In any triangle, when you extend one of its sides, you create an exterior angle. This exterior angle has a special relationship with the two interior angles that are not adjacent to it. The measure of the exterior angle is indeed equal to the sum of the measures of the two opposite interior angles. This […]
How do you simplify the expression 3x² + 3 + 9x³ + 4x³ + 2x² + 16?
To simplify the expression 3x² + 3 + 9x³ + 4x³ + 2x² + 16, we start by combining like terms. First, let’s organize the expression: Terms with x³: 9x³ + 4x³ Terms with x²: 3x² + 2x² Constant terms: 3 + 16 Next, we combine the like terms: x³: 9x³ + 4x³ = 13x³ […]
The sum of three consecutive multiples of 8 is 888. Find the multiples.
To solve the problem, we need to find three consecutive multiples of 8 that add up to 888. Let’s denote the first multiple of 8 as x. Then the next two consecutive multiples can be expressed as: First multiple: x Second multiple: x + 8 Third multiple: x + 16 Now, we can set up […]