Modeling #3 - Questions and Answers

Question 1 Reciprocal functions are useful for modeling situations where a constant amount is divided by one variable to produce the other variable. Make up a situation like this, then define the variables and write a reciprocal function that describes it. Answer Let C = the constant amount. In the reciprocal function we would have f(x) = C/x C could be the cost to hire a clown or anything else, x would be the number of people that will share the cost and f(x) would the cost divided evenly among them. The more people there are the lower the cost would be for each person. Question 2 Write and solve an equation to determine the width w of the flag [9.5ft] The length of an American flat is 1.9x its width Equation = 9.5 The width of the flag is feet. Answer 1 Given that an American flag's length is 1.9 times its width, the length is expressed as 1.9w. So, the equation is: 1.9w = 9.5 Divide both sides by 1.9 to find w: w = 5 So, the width of the flag is 5 feet. Answer 2 The equation representing the relationship between the length and width of the flag is: 1.9w = 9.5 To solve for the width, we need to isolate w on one side of the equation. We can do this by dividing both sides of the equation by 1.9: 1.9w / 1.9 = 9.5 / 1.9 Simplifying the equation gives: w = 9.5/1.9 Therefore, the width of the flag is 9.5 / 1.9 feet, or approximately 5 feet. Answer 3 The problem states that the length of the flag is 1.9 times its width. Using this information, we can write the equation: {Length of the flag} = 1.9 × {Width of the flag} Given that the length of the flag is 9.5 feet, we can substitute this value into the equation: 9.5 = 1.9w To solve for w, we need to isolate the variable. We can do this by dividing both sides of the equation by 1.9: 9.5 / 1.9 = 1.9w / 1.9 Simplifying the right side: 9.5 / 1.9 = w Performing the division: w = 5 Therefore, the width of the flag is 5 feet. Answer 4 Result: 5 = {Width} Solution: The length of an American flag is 1.9 times its width, so we can express this relationship as: {Length} = 1.9 × {Width} We are given that the length is 9.5 feet, so we can substitute this value into the equation: 9.5 = 1.9 × {Width} To solve for the width {Width}, we need to isolate it on one side of the equation. We can divide both sides of the equation by 1.9: 9.5 / 1.9 = (1.9 * {Width}) / 1.9 Simplifying the right side of the equation gives: 9.5 / 1.9 = {Width} Evaluating the division gives: 5 = {Width} Therefore, the width of the flag is 5 feet. Question 3 Which of the following statements is/are correct about logistic regression? Logistic regression can be used for modeling the continuous response variable with dichotomous explanatory variable. Logistic regression can be used for modeling the dichotomous response variable with dichotomous explanatory variable. Logistic regression can be used for modeling the continuous response variable with dichotomous or other type of categorical explanatory variables. Logistic regression can be used for modeling the dichotomous response variable with dichotomous and not for continuous explanatory variables. Logistic regression can be used for modeling the dichotomous response variable with categorical explanatory variables and/or continuous explanatory variables. Answer For logistic regression, the explanatory variables can either be categorical or continuous but the output has to be dichotomous. Hence, E. Logistic regression can be used for modeling the dichotomous response variable with categorical explanatory variables and/or continuous explanatory variables. Question 4 Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip? Answer Let's note: x=the number of bags of cement that can be safely lifted on the elevator in the same trip. We have: 245 + 95x ≤ 3000 We solve the inequality: 245 + 95x - 245 ≤ 3000 - 245 95x ≤ 2755 x ≤ 2755/95 x ≤ 29 Question 5 The stature of men is normally distributed, with a mean of 69.0 inches and a standard deviation of 2.8 inches. The height of women is normally distributed, with a mean of 63.6 inches and a standard deviation of 2.5 inches. Modeling academy standards require women to be models taller than 66 inches (or 5 feet 6 inches). What percentage of women meet this requirement? Answer Step 1 From the given information, the mean of women is 63.6 inches and standard deviation is 2.5 inches. Let X be the height of the women follows normal distribution with μ = 63.6 and σ = 2.5. Modeling academy standards require women to be models taller than 66 inches Step 2 The percentage of women meet this requirement is, Percentage = P(X > 66) × 100% = P((X-μ)/σ > (66 – μ)/σ) × 100% = P(z > (66 - 63.6) / 2.5) × 100% = P(z > 0.96) × 100% = [1 – P(z < 0.96)] × 100% = 0.1685 × 100% from the excel function, = 1 - NORM.DIST(0.96, 0, 1, TRUE) = 16.85%