Mathematics Answer Note #25

Question 1 Speedster Road Paving Company uses 5.7 pounds of cement to pave one meter of roadway. How many pounds of cement will the company use to pave 13 meters? Answer Based on the given information, we can determine that 5.7 pounds of cement is used to pave one meter of roadway by Speedster Road Paving Company. To calculate the amount of cement needed to pave 13 meters, we can use the following formula: Cement needed = Pounds of cement per meter * Number of meters to pave Therefore, we can plug in the values from the question into the formula as follows: Cement needed = 5.7 pounds/meter * 13 meters = 74.1 pounds Therefore, the company will need 74.1 pounds of cement to pave 13 meters of roadway. These sources provide useful guidelines and tools for accurately calculating the amount of concrete needed for a project, taking into account factors such as thickness and square footage. By utilizing these resources, we can ensure an accurate and efficient calculation of the cement needed for Speedster Road Paving Company to pave 13 meters of roadway. Additionally, these sources can also be used to calculate other material needs in road construction projects. As a college student, it is important to critically evaluate and utilize various academic sources to provide the most accurate and comprehensive response. Question 2 Compute the fourier series for the given function f on the specified integral Answer It is important to note that computing the Fourier series for a given function on a specified integral involves breaking down the function into its basic trigonometric components and representing it as a sum of sine and cosine waves. This process is used to analyze periodic functions and can be applied to a wide range of engineering and scientific problems. In order to compute the Fourier series for a given function, we first need to identify the period of the function. The period is defined as the smallest interval in which the function repeats itself. Once the period is determined, we can then express the given function as a series of sines and cosines and calculate the corresponding coefficients. To calculate the Fourier coefficients, we use the following equations: $a_0 = frac {1}{T} int_{-T/2}^{T/2} f(x) dx$ $a_n = frac {2}{T} int_{-T/2}^{T/2} f(x)cos(frac{2pi n x}{T}) dx$ $b_n = frac {2}{T} int_{-T/2}^{T/2} f(x)sin(frac{2pi n x}{T}) dx$ Where $T$ is the period of the function and $n$ is the frequency of the sine or cosine wave. To illustrate this process, let us consider the function $f(x) = x$ defined on the interval $[-2,2]$. The period of this function is $4$ since it repeats itself every $4$ units. The Fourier coefficients can be calculated as follows: $a_0 = frac {1}{4} int_{-2}^{2} x dx = 0$ $a_n = frac{2}{4} int_{-2}^{2} xcos(frac{2pi nx}{4}) dx = frac{1}{pi n} sin(pi n)$ $b_n = frac{2}{4} int_{-2}^{2} xsin(frac{2pi nx}{4}) dx = frac{2}{pi} (-1)^n + frac{1}{pi n} cos(pi n)$ Therefore, the Fourier series of $f(x)$ on the interval $[-2,2]$ can be written as: $f(x) = sum_{n=1}^infty left[frac{2}{pi(-1)^n n} sin(frac{n pi}{2}) cos(frac{n pi x}{2}) + frac{1}{pi n^2} cos(frac{n pi}{2}) sin(frac{n pi x}{2})right]$ For a more comprehensive understanding and detailed explanation of Fourier series, I would recommend referencing the following sources: 1. "Fourier Series and Boundary Value Problems" by James Brown and Ruel Churchill 2. "Introduction to Fourier Analysis and Generalized Functions" by M. Rahman 3. "Fourier Analysis: An Introduction" by Elias Stein and Rami Shakarchi In conclusion, computing the Fourier series for a given function on a specified integral involves identifying the period of the function, calculating the corresponding coefficients using trigonometric formulas, and representing the function as a sum of sines and cosines. This process is fundamental in understanding periodic functions and plays a crucial role in various applications in engineering and science. Question 3 A cafeteria used 144.9 kilograms of beans to make 4 batches of chili. To the nearest tenth of a kilogram, what quantity of beans went into each one? Answer Based on the given information, we can calculate the quantity of beans in each batch of chili by dividing the total amount used (144.9 kilograms) by the number of batches made (4). This yields an answer of 36.225 kilograms per batch. Step-by-Step Calculation: 1. Divide the total amount of beans used (144.9 kilograms) by the number of batches made (4). 2. 144.9 kilograms / 4 batches = 36.225 kilograms per batch. Therefore, to the nearest tenth of a kilogram, each batch of chili required approximately 36.2 kilograms of beans. Question 4 What is the equation of the line that passes through the point left parenthesis, 5, comma, 8, right parenthesis(5,8) and has a slope of start fraction, 3, divided by, 5, end fraction 5 3 ? Answer To find the equation of a line, we will use the general form y = mx + b, where m represents the slope and b represents the y-intercept. In this case, we are given that the slope is 3/5 and the point (5,8) lies on the line. Step 1: Finding the y-intercept (b) To find the y-intercept, we can use the formula (x1, y1) and substitute the values of the given point (5,8) in the equation y = mx + b. 8 = (3/5)(5) + b 8=3+b b=8-3 b=5 Step 2: Writing the equation Now that we have the value of b, we can write the equation of the line. y = (3/5)x + 5 Therefore, the equation of the line that passes through the point (5,8) and has a slope of 3/5 is y = (3/5)x + 5. Question 5 What is the equation of the line that passes through the point left parenthesis, 5, comma, minus, 2, right parenthesis(5,−2) and has a slope of minus, start fraction, 2, divided by, 5, end fraction− 5 2 ? Answer To answer this question, we first need to understand the slope-intercept form of a line, which is y=mx+b, where m is the slope and b is the y-intercept. We are given the slope (2/5) and a point on the line (5,-2), so we can use the point-slope form of a line to find the equation. Point-slope form: y-y1 = m(x-x1) Where (x1,y1) is the given point and m is the slope. Substituting the values from our given point and slope into the formula, we get: y-(-2) = -2/5 (x-5) Simplifying, we get: y+2 = -2/5x +2 To convert this to slope-intercept form, we need to isolate y on one side of the equation. We can do this by subtracting 2 from both sides. y = -2/5x +2 -2 Simplifying, we get: y = -2/5x So, the equation of the line that passes through the point (5,-2) and has a slope of -2/5, is y = -2/5x.