What is the solution of log2x 416 4?
To solve the equation log2(x) = 416 + 4, we first simplify the right-hand side: log2(x) = 420 Next, we need to rewrite this logarithmic equation in its exponential form. The definition of a logarithm tells us that if logb(a) = c, then bc = a. Here, we have: 2420 = x Therefore, the solution […]
Circle L is Inscribed with Triangle GJK, Line Segment GJ is a Diameter
In this scenario, we have circle L, which is inscribed within triangle GJK. The line segment GJ serves as a diameter of the circle. This setup has some interesting geometric properties. Since GJ is a diameter of the circle, it implies that angle GJK, which is inscribed in the circle, must be a right angle […]
What is the Probability of the Complement of Event E?
To find the probability of the complement of an event, you can use the following formula: P(not E) = 1 – P(E) Given that the probability of event E is 0.31, we can substitute this value into the formula: P(not E) = 1 – 0.31 = 0.69 Therefore, the probability of the complement of event […]
When the coordinates 1 1 2 3 5 3 and 4 1 are joined, which shape is formed?
To determine the shape formed by the coordinates (1, 1), (2, 3), (5, 3), and (4, 1), we can visualize or plot these points on a graph. 1. **Plot the Points:** Start by plotting each point on the Cartesian coordinate system: (1, 1) (2, 3) (5, 3) (4, 1) 2. **Join the Points:** Connect the […]
How do you simplify the expression 3w² + 5w + 6 + 6w² + 4w + 2?
To simplify the expression, we first need to combine like terms. The given expression is: 3w² + 5w + 6 + 6w² + 4w + 2 Let’s group the like terms: For the w² terms: 3w² + 6w² = 9w² For the w terms: 5w + 4w = 9w For the constant terms: 6 + […]
Prove 3n ≤ n! by induction using a base of n = 3
To prove the inequality 3n ≤ n! for all integers n ≥ 3 using mathematical induction, we will follow these steps: Base Case: First, we need to verify the base case with n = 3: Left side: 3n = 3 × 3 = 9 Right side: n! = 3! = 3 × 2 × 1 […]
Solve 13 lnx 23 12 ln x lnx2 3x 22
To solve the expression 13 ln x + 23 – 12 ln x + ln(x^2) + 3x – 22, we start by simplifying the terms that involve natural logarithms and constant values. First, let’s combine the logarithmic terms: 13 ln x – 12 ln x + ln(x^2) = (13 – 12) ln x + ln(x^2) […]
What are the zeros of the polynomial function f(x) = x³ + x² – 20x?
To find the zeros of the polynomial function f(x) = x³ + x² – 20x, we need to set the function equal to zero and solve for x: x³ + x² – 20x = 0 We can factor out the common term, which in this case is x: x(x² + x – 20) = 0 […]
Find the Length of an Arc of an 8 Radius Circle if the Arc Measures 45 Degrees
To find the length of an arc in a circle, you can use the formula: Arc Length = (θ / 360) × (2πr) Where: θ is the angle in degrees (in this case, 45 degrees) r is the radius of the circle (in this case, 8) Now, plug in the values: Arc Length = (45 […]
What quadratic function represents the parabola with vertex (2, 20) and y-intercept (0, 12)?
To find the quadratic function that represents the parabola with a given vertex and y-intercept, we can use the vertex form of a quadratic equation, which is: f(x) = a(x – h)² + k In this equation, (h, k) is the vertex of the parabola. Here, the vertex given is (2, 20), so we have: […]