How to Find the Lateral Area of a Cylinder with Radius 4 and Height 5?
To find the lateral area of a cylinder, you can use the formula: Lateral Area = 2 × π × r × h Here, r is the radius and h is the height of the cylinder. In our case, the radius (r) is 4 and the height (h) is 5. Now let’s plug in the […]
Which of the following pairs is coprime: a) 13 and 14 b) 8 and 20 c) 31 and 59 d) 34 and 85?
To determine which pair of numbers is coprime, we need to find if their greatest common divisor (GCD) is 1. Let’s analyze each pair: a) 13 and 14: The GCD of 13 and 14 is 1 since 13 is a prime number and does not divide 14. Thus, 13 and 14 are coprime. b) 8 […]
Derive the equation of the parabola with a focus at (0, 1) and a directrix of y = 1
The parabola opens upwards, with its vertex at the point (0, 1). The standard form of the equation of a parabola that opens upwards is given by: x2 = 4p(y – k), where (h, k) = (0, 1) is the vertex of the parabola, and p is the distance from the vertex to the focus. […]
What is the value of y if the segment outside the circle is tangent to the circle?
To find the value of y when a segment outside the circle is tangent to the circle, we can use the property of tangents. A tangent to a circle is a line that touches the circle at exactly one point. Let’s assume we have a circle with center O and a point A outside the […]
What Are the Zeros of the Polynomial Function f(x) = x³ – 9x² – 20x?
To find the zeros of the polynomial function f(x) = x³ – 9x² – 20x, we need to set the function equal to zero: f(x) = x³ – 9x² – 20x = 0 First, we can factor out the common term, which in this case is x: x(x² – 9x – 20) = 0 This […]
What is the correct sum of the polynomials 3x³ + 5x + 8 and 5x³ + 7x + 3?
To find the sum of the given polynomials, we need to combine like terms from both expressions. The first polynomial is: 3x³ + 5x + 8 The second polynomial is: 5x³ + 7x + 3 Now, let’s arrange the terms by their degree: For the cubic terms: 3x³ + 5x³ results in 8x³. For the […]
What is the integral of sec x?
The integral of sec x is given by the formula: ∫ sec x dx = ln |sec x + tan x| + C where C is the constant of integration. To derive this result, we can use a clever trick that involves multiplying and dividing by (sec x + tan x): Start with the integral: […]
What is the slope of the line passing through (1, 2) and (3, 8)?
To find the slope of a line that passes through two points, we use the formula: slope (m) = (y2 – y1) / (x2 – x1) Here, we have two points: (1, 2) and (3, 8). In this case: (x1, y1) = (1, 2) (x2, y2) = (3, 8) Now, we can plug in the […]
If a triangle has a height of 14 inches and a base of 9 inches, what is its area?
The area of a triangle can be calculated using the formula: Area = (base × height) / 2 In this case, the height is 14 inches and the base is 9 inches. Plugging these values into the formula: Area = (9 inches × 14 inches) / 2 Calculating this gives: Area = 126 inches² / […]
Which of the following is a solution of x² + 10x + 36?
To find the solutions of the equation x² + 10x + 36 = 0, we can use the quadratic formula: x = (-b ± √(b² – 4ac)) / 2a In our equation, a = 1, b = 10, and c = 36. First, we calculate the discriminant: Discriminant (D) = b² – 4ac = 10² […]