The larger of two consecutive integers is 7 greater than twice the smaller. Find the integers.
Let’s denote the smaller integer as x. Since the integers are consecutive, the larger integer can be represented as x + 1. According to the problem, the larger integer is 7 greater than twice the smaller integer. We can translate this into the equation: x + 1 = 2x + 7 Now, let’s solve for […]
How to Find the Length of the Curve given by r(t) = 5t, 3 cos(t), 3 sin(t), 5t^5?
To find the length of the curve defined by the vector function r(t) = (5t, 3 cos(t), 3 sin(t), 5t^5), we use the formula for the arc length of a space curve: L = ∫ab ||r'(t)|| dt Here, ||r'(t)|| represents the magnitude of the derivative of the vector function. To proceed, we’ll first differentiate r(t): […]
The sum of two numbers is 40; one number is 4 times as large as the other. What are the numbers?
To solve this problem, we need to set up a couple of equations based on the information given. Let’s call the smaller number x. According to the problem, the larger number is 4 times as large as the smaller number, so we can express the larger number as 4x. Next, we know that the sum […]
How do you simplify the expression 2 20 3 7 2 5 4 63?
To simplify the expression 2 20 3 7 2 5 4 63, we first need to identify the mathematical operations we should perform. However, as it stands, this expression is not clear because it lacks operators (like +, -, *, /) between the numbers. Assuming you meant to find the product of these numbers, we […]
What Does Rotation Mean in Geometry?
In geometry, rotation refers to the circular movement of an object around a center point or axis. When we rotate a shape, we turn it to a certain angle in a specific direction. For example, if you have a triangle and you rotate it 90 degrees clockwise around a point, each vertex of the triangle […]
How can I simplify the expression 6x² + 11x + 3 + 2x² + 17x + 4?
To simplify the expression 6x² + 11x + 3 + 2x² + 17x + 4, we will combine like terms. First, we identify the like terms: The terms with x²: 6x² and 2x² The terms with x: 11x and 17x The constant terms: 3 and 4 Next, we add the coefficients of the like terms: […]
Which Expression Does Not Factor?
When we talk about factoring expressions, we’re referring to the process of breaking down a polynomial into simpler components, or factors, that when multiplied together yield the original expression. However, some expressions cannot be factored using real numbers or standard algebraic methods. For example, the expression x² + 1 does not factor over the real […]
Which of the following statements is true for the following data set: 7, 5, 6, 4, 7, 8, 12?
To analyze the data set 7, 5, 6, 4, 7, 8, 12, let’s first calculate some key statistics: the mean, median, mode, and range. Mean: To find the mean, add all the numbers together and divide by the count of numbers. Mean = (7 + 5 + 6 + 4 + 7 + 8 + […]
Fill in the Blanks: 1. Zero has Reciprocal 2. The Numbers ___ and ___ are Their Own Reciprocals
1. Zero has no reciprocal. Explanation: The reciprocal of a number is defined as 1 divided by that number. However, since dividing by zero is undefined in mathematics, zero does not have a reciprocal. This is a fundamental concept in arithmetic. 2. The numbers 1 and -1 are their own reciprocals. Explanation: A number is […]
What is the value of f(x) = 2x for f(2)?
To find the value of f(2) for the function f(x) = 2x, we need to substitute x with 2 in the equation. The calculation goes as follows: f(2) = 2 * 2 = 4 Therefore, the value of f(2) is 4. This means that when we plug in 2 into the function, we simply multiply […]