How to find dy/dx using the equation 4y cos(x) + x^2y^2?
To find dy/dx from the equation 4y cos(x) + x2y2 = 0, we will use implicit differentiation. First, let’s differentiate both sides of the equation with respect to x: d/dx (4y cos(x) + x2y2) = 0 Now, apply the product rule and chain rule as necessary: For the term 4y cos(x): Using the product rule: […]
Find the Exact Length of the Curve: x = 1/3y^3 + 16y + 25
To find the exact length of the curve given by the equation x = (1/3)y^3 + 16y + 25, we will use the formula for the length of a curve defined by a function: L = ∫ab √(1 + (dy/dx)²) dy First, we need to express everything in terms of y
How do you factorize the expression 5x³ + 135 + 5x + 3x² + 9?
To factorize the expression 5x³ + 135 + 5x + 3x² + 9, we will start by grouping and simplifying the terms. The expression can be rearranged for clarity: 5x³ + 3x² + 5x + 135 + 9 Next, we can try to group the terms: (5x³ + 3x² + 5x) + (135 + 9) […]
What must be subtracted from 3 to get 9?
To find out what must be subtracted from 3 to get 9, we can set up the equation: 3 – x = 9 Here, x represents the number that we need to find. To isolate x, we can rearrange the equation: -x = 9 – 3 -x = 6 Now, to solve for x, we […]
Which of the following is the equation of a parabola with focus (0, 2) and directrix y = 2?
To determine the equation of a parabola given its focus and directrix, we start by identifying the components. The focus of the parabola is the point where all the rays converge, which in this case is (0, 2). The directrix is a line, which for our problem is given by y = 2. In a […]
How do you solve for x in the equation √(x + 9) = 41?
To solve the equation √(x + 9) = 41, we need to isolate x. Let’s go through the steps together: Start with the original equation: √(x + 9) = 41. To eliminate the square root, we can square both sides of the equation. This gives us: (√(x + 9))² = 41². Simplifying both sides leads […]
What is the solution of the system of equations y = 4x + 4 and y = 3x + 3?
To find the solution of the given system of equations, we need to find the values of x and y that satisfy both equations simultaneously. We have: 1. y = 4x + 4 2. y = 3x + 3 Since both equations equal y, we can set them equal to each other: 4x + 4 […]
In the rhombus angle 1 is 140. What are angles 2 and 3? The diagram is not to scale.
In a rhombus, opposite angles are equal, and adjacent angles are supplementary. Since angle 1 is given as 140 degrees, the opposite angle (angle 3) will also be 140 degrees. To find angle 2, we need to remember that adjacent angles in a rhombus add up to 180 degrees. Therefore, we can calculate angle 2 […]
Which is a factor of 6x³y, 6xy, 12x², 12, x², 1, 2?
To determine which expressions are factors of the other terms given (6x³y, 6xy, 12x², 12, x², 1, and 2), we need to analyze each one. 1. **Factors of 6x³y**: This expression is divisible by: 6xy (it can be simplified by dividing by x² and y) 6xy is also a factor of itself. 12x² is not […]
What is the equation of the horizontal line that passes through the point (2, 3)?
A horizontal line is defined by a constant y-value. Since the line passes through the point (2, 3), we can see that the y-coordinate is 3. Therefore, the equation of the horizontal line is given by: y = 3 This means that no matter what the x-coordinate is, the value of y will always be […]