What is the radius of a circle whose equation is x^2 + y^2 = 42?
The equation of a circle in the standard form is given by: (x – h)² + (y – k)² = r² Here, (h, k) is the center of the circle and r is the radius. If the equation is written as: x² + y² = r² then the circle is centered at the origin (0, […]
What is the slope of the line tangent to the graph of y = ln(2x) at the point where x = 4?
To find the slope of the line tangent to the graph of y = ln(2x) at the point where x = 4, we first need to calculate the derivative of the function with respect to x. The derivative of y = ln(2x) can be found using the chain rule. The derivative is: dy/dx = (1/2x) […]
What exponential function represents the data in the table x f(x) 3: 64, 4: 256, 5: 1024?
To find the exponential function that represents the given data, we will analyze the values provided in the table. The data points are: x = 3, f(x) = 64 x = 4, f(x) = 256 x = 5, f(x) = 1024 We can note that the values of f(x) increase significantly as x increases by […]
How to Find the Exact Length of the Polar Curve r = 8 + 8cos(θ)?
To find the length of the polar curve given by the equation r = 8 + 8cos(θ), we use the formula for the length of a polar curve: L = ∫αβ √(r² + (dr/dθ)²) dθ Here, the first step is to determine the values of α and β, which are the angles that define the […]
Given the following linear function, sketch the graph of the function and find the domain and range: f(x) = 5x + 4
The function f(x) = 5x + 4 is a linear function, which means its graph is a straight line. To sketch the graph, we can find two points by choosing different values for x. Let’s find the y-intercept (where the line crosses the y-axis) by setting x to 0: f(0) = 5(0) + 4 = […]
In the Regular Nonagon Shown, What is the Measure of Angle X?
To find the measure of angle X in a regular nonagon, we need to use the formula for calculating the interior angle of a regular polygon. The formula for the measure of each interior angle in a regular polygon is: Interior Angle = (n – 2) × 180° / n where n is the number […]
What is the midpoint of the x intercepts of f(x) = x^2 + x – 4?
To find the midpoint of the x intercepts of the function f(x) = x2 + x – 4, we first need to determine the x intercepts themselves. The x intercepts occur where the function equals zero, so we set the equation to zero: x2 + x – 4 = 0 Next, we can use the […]
Which system of linear inequalities has the point (3, 2) in its solution set?
To determine which system of linear inequalities includes the point (3, 2) in its solution set, we need to check the inequalities against this point. Start by considering a couple of example inequalities. Let’s say we have the following system: 1. y > 2x – 4 2. y < -x + 5 Now, we will […]
What is the exact length of arc ABC in circle E with a radius of 40 centimeters and an arc measure of 324 degrees?
To find the exact length of arc ABC, we can use the formula for the length of an arc, which is given by: Arc Length = (θ / 360) * 2πr Where: θ is the angle in degrees (in this case, 324 degrees), r is the radius of the circle (40 centimeters), π is approximately […]
What are the zeros of the polynomial function f(x) = x³ + x² – 12x?
To find the zeros of the polynomial function f(x) = x³ + x² – 12x, we first set the function equal to zero: x³ + x² – 12x = 0 Next, we can factor the equation. Notice that there is a common factor of x in each term: x(x² + x – 12) = 0 […]