To solve for the area of the walk surrounding the circular fountain, you need to first calculate the area of the fountain itself and then the area of the larger circle that includes the walk. 1. **Calculate the radius of the fountain**: The fountain has a diameter of 10 feet, so the radius (r) of … Read more
To find the 7th term of a geometric sequence, we first need to identify the common ratio and use it to determine the 7th term. In a geometric sequence, each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio (often denoted as r). The general formula for … Read more
To find the equation of the tangent line to the curve at a specific point, we need to follow a systematic approach. In this case, we are given the curve: y = x3 + 2x + 1 and the point at which we want to find the tangent line: (3, 22). Step 1: Differentiate the … Read more
To find the derivative of the function g(x) = 8x using the definition of the derivative, we start by applying the limit definition: Definition of the Derivative: The derivative of a function g(x) at a point x is defined as: g'(x) = lim (h → 0) [(g(x + h) – g(x)) / h] Now, let’s … Read more
To solve the integral of the function sin(x) cos(x) + 3sin^2(x), we can break it down into two parts: sin(x) cos(x) and 3sin^2(x). Step 1: Integrating sin(x) cos(x) The integral of sin(x) cos(x) can be simplified using a trigonometric identity: sin(x) cos(x) = (1/2) sin(2x) Therefore, we have: ∫sin(x) cos(x) dx = ∫(1/2) sin(2x) dx … Read more
Converting a fraction to a mixed number involves a few simple steps. A mixed number consists of an integer and a proper fraction combined. Here’s how you can do it: Identify the Fraction: Start with a fraction, for example, 7/4. Divide the Numerator by the Denominator: Perform the division. For our example, 7 ÷ 4 … Read more
To solve the problem, let’s define the two numbers: – Let X be the first number. – Let Y be the second number. According to the problem, if we subtract 40 percent of Y from Y, the result is 35. We can express this mathematically: Y – 0.4Y = 35 This simplifies to: 0.6Y = … Read more
To find the vertical asymptotes of the function \( f(x) = \frac{10}{x^2 – 7x – 30} \), we need to identify where the denominator equals zero, as vertical asymptotes occur at these values. 1. **Set the Denominator to Zero:** We begin by solving the equation: \( x^2 – 7x – 30 = 0 \) 2. … Read more
To determine which expression is a factor of 2xy + 6x + 3y + 9, we first need to look for common factors among the terms. Let’s break down each term: The first term is 2xy
To find a unit vector that has the same direction as the vector 3i + 7j, we can follow these steps: Step 1: Identify the vector components – The vector can be expressed as v = (3, 7), where its components are 3 and 7. Step 2: Calculate the magnitude of the vector – The … Read more