Identify the vertex and the axis of symmetry of the graph of the function y = 2x^2 + 2x + 4
To find the vertex and the axis of symmetry of the quadratic function y = 2x^2 + 2x + 4, we can use the vertex formula. The vertex of a parabola in the form y = ax^2 + bx + c can be found at the point (h, k), where: h = -b / (2a) […]
What is the quotient of x³ + 3x² + 5x + 3 by x + 1?
To find the quotient of the polynomial x³ + 3x² + 5x + 3 by x + 1, we can use polynomial long division. 1. Start by dividing the first term of the dividend (x³) by the first term of the divisor (x). This gives us x². 2. Multiply x² by the entire divisor (x […]
What is the volume of a cube if its surface area is 2400 cm²?
To find the volume of a cube when its surface area is given, we first need to understand the relationship between the surface area and the side length of the cube. The formula for the surface area (SA) of a cube is: SA = 6s² where s is the length of one side of the […]
Find All Solutions to the Equation sin x cos x = 0
To solve the equation sin x cos x = 0, we can utilize the property that the product of two terms equals zero when at least one of the terms is zero. This gives us two separate equations to consider: sin x = 0 cos x = 0 Let’s solve each of these cases: 1. […]
What is the simplified form of the fourth root of x squared times the fourth root of x squared?
To simplify the expression √²x⁴ * √²x⁴, we first rewrite it using exponent notation. The fourth root of x squared can be expressed as x raised to the power of 2/4, which simplifies to x raised to the power of 1/2 or √x. So, we have: √²x⁴ * √²x⁴ = (x^(2/4)) * (x^(2/4)) = x^(2/4 […]
Where is the CG of a scalene triangle and a cylinder?
The center of gravity (CG) of any object is the point at which the entire weight of the object can be considered to act. It’s crucial for understanding balance and stability in various structures. 1. Center of Gravity of a Scalene Triangle The CG of a scalene triangle, which has sides of unequal lengths and […]
How to Find the Least Squares Line for Given Data Points?
To find the equation of the least squares line that best fits the data points (2, 3), (3, 2), (5, 1), (6, 0), we can use the least squares method. This involves finding the slope (b) and intercept (a) of the line represented by the equation y = a + bx. First, let’s list our […]
What is the maximum height attained by an object with the height equation h = 10t – 5t²?
To find the maximum height attained by the object, we first need to analyze the height function: h(t) = 10t – 5t² This is a quadratic equation where a = -5, b = 10, and c = 0. The maximum point of a quadratic equation occurs at the vertex, which can be found using the […]
In a baseball game, a batter hits a 0.150 kg ball straight back at the pitcher at 180 km/h. What is the magnitude of the average force exerted by the bat on it if the collision lasts 5.0 ms?
To find the average force exerted by the bat on the ball, we can use the impulse-momentum theorem, which states that the impulse (force multiplied by the time the force acts) is equal to the change in momentum of the object. First, we need to convert the speed of the baseball from kilometers per hour […]
Find a unit vector that has the same direction as the given vector 8i + j + 4k
To find a unit vector in the same direction as a given vector, we first need to determine the magnitude of the vector. The given vector is 8i + j + 4k. The magnitude (or length) of a vector v = ai + bj + ck is calculated using the formula: $$ |v| = \sqrt{a^2 […]