GENERAL MATH

PLEASE EXPLAIN HOW'D YOU GET THE ANSWER IN THE SUPPLEMENTARY EXERCISE. HELP ME UNDERSTAND PLEASE.

lesson:

Lesson 4: Representing Real-Life Situations Using Rational Functions Learning Dutcomefs}: At the end of the lesson. the learner is able to represent real-life situations rational functions. Lesson Outline: 1. Review: Polynomial functions 2. Rational functions and real-life situations Recall the definition of a poivnomial function. Definition: A polynomial function p of degree n is a function that can be written in the form px} = (1,an + on_11"'1 + a,l_3x"'2 + + at: + on where \"13.51. ...,a,l E lit, an at [hand n is a positive integer. Each addend of the sum is a term of the polynomial function. The constants u:1.z-u--n are the coefficients. The leading coefficient is o... The leading term is ux"- and the constant term is u- Definition: A rational function is a function of the form fix) = E where pix] and QUE] are polynomial functions and '10:] is not the zero function {i.e.. tit-1') i D]. The domain of x) is the set of all values of x where qijx) a [1. Example 1. An object is to travel a distance of 1i] meters. Express veloch v as a function of travel time t. in seconds. Sofutfon. The following table of values show v for various values of t. v {meters per second) The function ctr} = % can represent v as a function of t. Example 2. Suppose that 50:} = {3:1 {in mgi'mL] represents the concentration of a drug in a patient's bloodstream t hours after the drug was administered. Construct a table of values for cm for t = 1.2. 5. 11]. Round off answers to three decimal places. Use the table to sketch a graph and interpret the results. Sofutr'on. J' = Flt) t...) -2-10123456?Elf The graph indicates that the maximum drug concentration occurs around 1 hour after the drug was administered [calculus can he used to determine the exact value at which the maximum occurs}. After \"I hour. the graph suggests that drug concentration decreases until it is almost zero. Solved Examples 1. In an organ pipe. the frequency f of vibration of air is inverselyr proportional to the length L of the pipe.1 Suppose that the frequency;r of vibration in a 10-foot pipe is 54 vibrations per second. Express fas a function of L. Solution. Since f is inverser proportional to L. then f = E where k is the constant of proportionality. If L = 1o then r = 54. Thus, st==a k = 54o. Thus: the function {[L} :1\" 5 1. represents f as a function of L. 2. The distance from Manila to Baguio is around 250 kilometers. la] How long will it take you to get to Baguio if your average speed is 25 kilometers per hour? 40 kilometers per hour? 50 kilometers per hour? lb} Construct a function [5) , where s is the speed of travel. that describes the time it takes to drive from Manila to Baguio. Solution. (a) Distance is calculated as the product of speed and time. So we can get the time by dividing distance by the speed. 250 kilometers/ 25 kilometers per hour = 10 hours 250 kilometers/ 40 kilometers per hour = 6.25 hours 250 kilometers/ 50 kilometers per hour = 5 hours (b) Since time is the quotient of distance and speed, we can write out the function as t(s) = The distance is fixed at 250 kilometers so the final function we have is 250 t()= S Lesson 4 Supplementary Exercises 1. Given the polynomial function p(x) = 12 + 4x - 3x2 - x3, find (a) The degree of the polynomial (b) The leading coefficient (c) The constant term (d) The number of zeroes 2. The budget of a university organization is split evenly among its various committees. If they have a budget of P60,000: (a) Construct a function M(n) which would give the amount of money each of the n number of committees would receive. (b) If the organization has eight committees, how much would each committee have? 3. A company has a budget of P90,000 to be split evenly among its various offices. The marketing office of the company receives twice the amount of money than the other offices. (a) Given x as the number of offices in the company, construct a function f(x) which would give the amount of money each of the non-marketing offices would receive. (b) If the company had five offices, how much would the marketing office receive? How much would each of the non-marketing offices receive? 4. Let C(t) = be the function that describes the concentration of a certain medication in the bloodstream over time t. (a) What is C(0)? Why is that so? (b) Construct a table of values for when t is equal to 0, 1,2,3,4, and 5. (c) Interpret your answers in relation to drug concentration