How to Write an Equation of an Ellipse in Standard Form with Given Vertices?
To find the equation of an ellipse in standard form with the center at the origin and given vertices, we first need to identify some essential characteristics based on the provided information. Here, the vertex is at (30, 0) and the co-vertex is at (0, 2). The standard form of an ellipse centered at the […]
What is the polynomial when the quotient of x^4 + 5x^3 + 3x + 15 is x^3 + 3?
To find the polynomial that the expression x^4 + 5x^3 + 3x + 15 is divided by to get the quotient x^3 + 3, we can use polynomial long division. 1. Start with the equation: x^4 + 5x^3 + 3x + 15 = (x^3 + 3) * P(x) 2. Here, P(x) is the polynomial we […]
What is the measure of an acute base angle of an isosceles trapezoid that is part of an isosceles triangle with a 34 degree vertex angle?
To find the measure of an acute base angle of the isosceles trapezoid that is part of an isosceles triangle with a vertex angle of 34 degrees, we can follow these steps: 1. **Understanding the Triangle**: First, let’s visualize the isosceles triangle. It has two equal sides, so the base angles are also equal. The […]
What value of x is in the solution set of 4x < 12 < 16 < 8x?
To find the value of x that satisfies the inequality 4x < 12 < 16 < 8x, we will break it down into two parts for clarity. First, let’s solve the inequality 4x < 12. To isolate x, divide both sides by 4: x < 3 Next, solve the inequality 8x > 16. Again, isolate […]
In triangle DEF, CG is 5 units and DG is 3x + 2. What is DG?
To find the value of DG, we need to express DG in terms of the variable x. The question states that DG is represented by the expression 3x + 2. However, we first need to determine the value for x. Given that CG is 5 units, we can deduce that if the problem provides a […]
What is the difference between a sequence and a series?
In mathematics, a sequence is a list of numbers arranged in a specific order. Each number in the sequence is called a term, and the sequence can be either finite or infinite. For example, the numbers 1, 2, 3, 4, and 5 can be expressed as a finite sequence, while the sequence of all natural […]
Find the value of the integral 1 2cosx 2 cosx2 dx from 0 to 2
To find the value of the integral ∫ (1 + 2cos(x) + 2cos(x2)) dx from 0 to 2, we can break it down into simpler parts. First, we split the integral: ∫ (1 + 2cos(x) + 2cos(x2)) dx = ∫ 1 dx + 2∫ cos(x) dx + 2∫ cos(x2) dx Now, we evaluate each part: […]
Which equation represents the line that passes through the points (6, 3) and (4, 9)?
To find the equation of the line that passes through the points (6, 3) and (4, 9), we start by determining the slope of the line using the formula: m = (y2 – y1) / (x2 – x1) Here, we can let (x1, y1) be (6, 3) and (x2, y2) be (4, 9). Substituting the […]
What is the volume of a cube shaped box with edges 6 centimeters in length?
The volume of a cube can be calculated using the formula: Volume = side × side × side or Volume = side³ In this case, the length of each edge of the cube is 6 centimeters. To find the volume, we simply plug this value into the formula: Volume = 6 cm × 6 cm […]
The sum of 3 numbers in AP is 18. If the product of the first and third number is 5 times the common difference, find the numbers.
Let the three numbers in Arithmetic Progression (AP) be: a – d, a, a + d where a is the middle term and d is the common difference. According to the problem, the sum of these three numbers is 18: (a – d) + a + (a + d) = 18 Simplifying gives: 3a = […]