To determine if the system of equations is independent, dependent, or inconsistent, we can analyze the equations algebraically without graphing. The two equations in the system are: Equation 1: 2x + y = 9 Equation 2: 3x + 4y = 8 We can start by expressing one variable in terms of the other using Equation … Read more
To find the largest area of a rectangle inscribed under the parabola given by the equation y = 12 – x² and with its base on the x-axis, we can follow these steps: First, we recognize that the rectangle has its base on the x-axis, and its upper corners will touch the parabola. Since the … Read more
Factoring the polynomial x³ + 11x² + 3x + 33 by grouping is a systematic approach that can simplify the process of determining its factors. Here’s a step-by-step guide on how to do it: Group the Terms: First, we will group the polynomial into two pairs. This can help us factor them separately. So, we … Read more
To solve this problem, we start with the concept of joint variation. The relationship can be expressed with the formula: x = k imes y imes z where k is the constant of variation. Step 1: Find the constant of variation (k) We know from the question that when x = 8, y = 4, … Read more
The probability of an event that is certain to occur is always 1, or 100%. In probability theory, probabilities range from 0 to 1, where 0 represents an impossible event, and 1 signifies an event that will definitely happen. To understand why certain events have a probability of 1, consider an example: the event of … Read more
To find the radius of a circle where the area and circumference are equal, we can start with the formulas for area and circumference: Area (A) = πr² Circumference (C) = 2πr We want to set these two equations equal to each other: A = C πr² = 2πr Next, we can simplify by dividing … Read more
To find the quotient of x³ divided by 8x², we need to simplify the expression. Here’s the step-by-step breakdown of the process: Write the expression: We start with the expression x³ ÷ 8x². Divide the coefficients: The coefficient of x³ is 1 (since it’s not explicitly written), and the coefficient of 8x² is 8. Therefore, … Read more
To find the area of a triangle, you can use the following formula: Area = (base × height) / 2 In this case, you have: Base = 9 inches Height = 14 inches Now, plug in the values: Area = (9 inches × 14 inches) / 2 Calculating this: Area = (126 inches²) / 2 … Read more
To evaluate the double integral d = ˆ{x^2 + 6y} dA, where the region of integration is bounded by y = x, y = x^3 and x = 0, we first need to determine the limits of integration by finding the points of intersection of the curves. 1. **Finding Points of Intersection:** To find where … Read more
The area of a trapezoid can be calculated using the following formula: Area = (1/2) * (b1 + b2) * h Where: b1: Length of the first base (the top side of the trapezoid). b2: Length of the second base (the bottom side of the trapezoid). h: Height of the trapezoid (the perpendicular distance between … Read more