Describe Parametric Equation of a Circle
The parametric equations for a circle are a set of equations that express the coordinates of points on the circle as functions of a variable, usually denoted as t. For a circle centered at the origin with radius r, the parametric equations can be expressed as: x(t) = r * cos(t) y(t) = r * […]
How do we find dy/dx and d²y/dx² for y = cos²(t) when t is in the interval [0, π]?
To find the first and second derivatives of the function y = cos²(t), we will use the chain rule for differentiation. Step 1: Finding dy/dx First, we need to differentiate y with respect to t: Using the chain rule: dy/dt = d(cos²(t))/dt We apply the derivative of cos²(t): dy/dt = 2cos(t) * (-sin(t)) = -2cos(t)sin(t) […]
Which of the following is the solution set of 18 – 3n < 2n + 20 - 4n?
To solve the inequality, we will first simplify it step by step. Starting with: 18 – 3n < 2n + 20 - 4n First, let’s combine like terms on the right side: 18 – 3n < (2n - 4n) + 20 This simplifies to: 18 – 3n < -2n + 20 Next, we want to […]
If you flip a fair coin 7 times, what is the probability that you will get exactly 2 tails?
To find the probability of getting exactly 2 tails when flipping a fair coin 7 times, we can use the binomial probability formula: P(X = k) = C(n, k) * p^k * (1-p)^(n-k) Where: C(n, k) is the number of combinations of n items taken k at a time, p is the probability of success […]
What does the diagram of the factorization of x² + 9x + 18 represent?
To start, the expression x² + 9x + 18 can be factored to find its roots. We’re looking for two numbers that multiply to 18 (the constant term) and add up to 9 (the coefficient of x). In this case, the numbers 3 and 6 fit perfectly: 3 x 6 = 18 and 3 + […]
How do I write the quadratic function in vertex form for y = x² + 4x + 7?
To convert the quadratic function y = x² + 4x + 7 into vertex form, we need to complete the square. Starting with the given equation: y = x² + 4x + 7 First, we focus on the x² + 4x part. To complete the square, we look for a number that, when added and […]
What is an exponential function for compounding interest when the rate of change is 5?
To express an exponential function for compounding interest, we can start with the general formula: A = P(1 + r/n)^(nt) In this formula: A is the amount of money accumulated after n years, including interest. P is the principal amount (the initial amount of money). r is the annual interest rate (decimal). n is the […]
If the sum of the interior angles of a polygon is 540, how many sides does the polygon have?
To determine the number of sides in a polygon based on the sum of its interior angles, we can use the formula: (n – 2) × 180 = Sum of interior angles where n is the number of sides in the polygon. In this case, we know that the sum of the interior angles is […]
How to Find dy/dx and d²y/dx² for y = t?
To find the first and second derivatives of y with respect to x for the equation y = t, we need to express t in terms of x. Since the problem does not provide a specific relationship between t and x, we will assume that t is a constant. Therefore, the value of y = […]
Which equation represents a line that passes through (9, 3) and has a slope of 6?
To find the equation of a line given a point and the slope, we can use the point-slope form of the equation of a line, which is: Y – Y1 = m(X – X1) In this formula, (X1, Y1) is a point on the line (in this case, (9, 3)), and m is the slope […]