How to Write the Equation of a Sphere in Standard Form: 2x² + 2y² + 2z² + 12x + 24z + 1
To convert the equation of the sphere given by 2x² + 2y² + 2z² + 12x + 24z + 1 = 0 into standard form, we need to follow these steps: Divide the equation by 2: This simplifies the coefficients of the squared terms: x² + y² + z² + 6x + 12z + rac{1}{2} […]
What are the possible dimensions of a rectangular room with an area given by the trinomial x² + 3x – 28?
To find the possible dimensions of the rectangle, we need to factor the trinomial x² + 3x – 28. The goal is to express this equation in the form of (x + a)(x + b) = 0, where a and b are constants. First, we look for two numbers that multiply to -28 (the constant […]
Express the area A of a circle as a function of its radius r, diameter d, and circumference C.
The area A of a circle can be expressed in different ways depending on whether we are using the radius, diameter, or circumference. 1. Area as a function of radius (r) The area A of a circle in terms of its radius r is given by the formula: A = πr² In this formula, π […]
How can we find the exact value of sin 255 using addition or subtraction formulas?
To find the exact value of sin 255 degrees, we can use addition or subtraction formulas from trigonometry. Specifically, we can express 255 degrees as a sum of two angles whose sine values we know. One convenient way to do this is to write: sin(255) = sin(270 – 15) Now we can apply the sine […]
What is the domain of the function f(x) = 2x + 5?
The domain of the function f(x) = 2x + 5 is all real numbers. This is because the function is a linear function, and linear functions do not have any restrictions on their inputs. You can substitute any real number for x, and you will get a valid output. In mathematical terms, this means that […]
How many terms are in the binomial expansion of 3x^59?
The number of terms in the binomial expansion of an expression of the form (a + b)n is given by the formula n + 1, where n is the exponent. In this case, we have the expression (3x + b)59. Here, n is equal to 59. Thus, to find the number of terms in the […]
Which Set Represents the Range of the Function Shown: 1, 5, 2, 8, 5, 3, 13, 4?
To determine the range of the function represented by the given set of numbers, we first need to identify the unique output values. The values provided are: 1, 5, 2, 8, 5, 3, 13, and 4. Next, we remove any duplicate values from this set. The number 5 appears twice, so we only list it […]
In each of the following pairs of numbers, state which whole number is on the left of the other number on the number line.
To determine which whole number is on the left of the other number on the number line, we need to understand that in a number line, numbers increase from left to right. This means that a smaller number will always be positioned to the left of a larger number. For instance, consider the pair of […]
Find the integral of sin(x) cos(x) 3sin(2x) dx
To solve the integral of sin(x) cos(x) 3sin(2x) dx, we start by simplifying the expression. We know that sin(2x) = 2sin(x)cos(x), thus 3sin(2x) = 6sin(x)cos(x). This means we can rewrite the integral as: ∫ sin(x) cos(x) 6sin(x) cos(x) dx Combining terms, we have: ∫ 6sin^2(x) cos^2(x) dx Next, we can use the identity for sin^2(x): […]
Identify the horizontal asymptote of f(x) = (7x + 1) / (2x – 9)
To determine the horizontal asymptote of the function f(x) = (7x + 1) / (2x – 9), we need to analyze the degrees of the polynomial in the numerator and the polynomial in the denominator. 1. **Identify the Degrees**: Both the numerator and the denominator are linear polynomials, meaning their highest degree is 1. 2. […]