Questions and Answers #6 Modeling

Questions and Answers Sheet 6 Modeling Question #1 The function represents the height y (in feet) of a grasshopper jumping straight up from the ground f seconds after the start of the jump. After how many seconds is the grasshopper 1 foot oft the ground? Given: Function 𝑦 = −16𝑡 2 + 8𝑡 represents the height (in feet) of grasshopper at t seconds. Answer: Calculations: The grasshopper is 1 foot off the ground when y=1 . So, 1 = −16𝑡 2 + 8𝑡 16𝑡 2 − 8𝑡 + 1 = 0 (4𝑡 − 1)2 = 0 1 𝑡= 4 Thus, required time is 1 4 seconds. Question #2 The function 𝑦 = 3.5𝑥 + 2.8 represents the cost y (in dollars) of a taxi ride of x miles. a. Identify the independent and dependent variables. b. You have enough money to travel at most 20 miles in the taxi. Find the domain and range of the function. Answer: a. Here, the independent variable is x and the dependent variable is y as the cost of taxi ride depends on the number of miles travelled. Independetn variable: x Dependent variable: y Question #3 The number of individual memberships at a fitness center in m months is represented by 142+12m. The number of family memberships at the fitness center in m months is represented by 52+6m. Write a polynomial that represents the total number of memberships at the fitness center. Answer: Given: Number of individual memberships at the fitness center in m months, say a =142+12m Number of family memberships at the fitness center in m months, say b =52+6m Calculation: Total number of memberships at the fitness center =a+b =142+12m+52+6m =18m+194 Thus, the polynomial representing the total number of memberships at the fitness center is 18m+194 . Question #4 The amount of money you hate after investing $400 for 8 years and $600 for 6 years at the same interest rate is represented by 400𝑥 8 + 600𝑥 6 where x is the growth factor. Classify the polynomial by the number of terms. What is the degree? Answer: Calculation: The given polynomial is 400𝑥 8 + 600𝑥 6 400𝑥 8 + 600𝑥 6 Have 2 terms. Therefore it is binomial. The maximum exponent of 400𝑥 8 + 600𝑥 6 is 8 Therefore degree is 8. Question #5 The temperature at 5 p.m. is 20∘ 𝐹 . The temperature at 19 p.m. is −5∘ 𝐹 . How many degree did the temperature fall? To solve: The temperature fall between 5PM to 10PM. Answer: To find the temperature fall, we will find the difference between both the temperatures. Suppose temperature fall is t. t=20-(-5) t=20+5 𝑡 = 25∘ 𝐹 Question #6 On a trip, the total distance (in miles) you travel in x hours is represented by the piecewise function. if 0  x  2 55x, d ( x) =  65x − 20, if 2  x  5 How far do you travel in 4 hours? Answer: Calculation: The given function: if 0  x  2 55x, d ( x) =  65x − 20, if 2  x  5 Use the second equation because 2<4<5 . d(x)=65x-20 d(4)=65(4)-20 d(4)=260-20 d(4)=240 Therefore the distance travel in 4 hours is 240 miles. Question #7 The temperature at 5 P.M. is 20∘ 𝐹. The temperature at 10 P.M. is −5∘ 𝐹. How many degrees did the temperature fall? Answer: Calculation: To find the temperature fall, we will find the difference between both the temperatures. Let the fall in temperature be x. 𝑡 = 20 − (−5) 𝑡 = 25∘ 𝐹 Conclusion: Therefore, the fall in temperature is 25∘ 𝐹. Question #8 To find: Whether you receive an A on the test. Given information: The inequality 3𝑥 + 2𝑦 ≥ 93 represents the number x of multiple choice questions and the numbers y of matching questions you can answer correctly to receive an A on a test. You answer 20 multiple choice question and 18 matching questions correctly. Answer: Calculation: The inequality 3𝑥 + 2𝑦 ≥ 93 indicates that the minimum number to receive an A on the test is 93. 20 multiple choice questions and 18 matching questions are answered correctly. Then, 3x+2y = 3 × 20 + 2 × 18 =60+36 =96 You scored a total of 96 which is greater than 93. Thus, you receive an A on the test. Question #9 To find: the volume of the sphere to the nearest cubic meter. Given: The surface area of the sphere is 𝑆 = 60𝑚2 Answer: Concept used: Volume of the sphere whose surface area is S is given by: 𝑉= 1 6√𝜋 3 𝑆2 Calculation: The volume of the sphere is: 𝑉= = ≈ ≈ 1 6√𝜋 1 6√3.14 1 3 𝑆2 3 (60)2 6(1.772) 1 10.632 [7.74597]3 (464.7586) ≈ 43.71 Conclusion: The volume of the sphere (to the nearest cubic meter) is about 44𝑚3 . Question #10 a) To write: A binomial that represents the height of the projection. Given information: Height of wall is xft. and width of projection is (x-3)ft. Area is represented by 𝑥 2 − 8𝑥 + 15. b) To find: Perimeter of projection when height of the wall is 8ft. Given information: Height of wall is x=8ft and width of projection is (x-3)ft. Height of projection is (x-5)ft. Answer: a) Calculation: Writing the left hand side factored and simplifying, 𝑥 2 − 8𝑥 + 15 = 𝑥 2 − 5𝑥 − 3𝑥 + 15 = 𝑥(𝑥 − 5) − 3(𝑥 − 5) = (𝑥 − 3)(𝑥 − 5) The height of the projection is represented by binomial (x-5) . b) Calculation: Width of projection =(x-3)ft Height of protection =(x-5)ft Perimeter of protection = 2[(𝑥 − 5) + (𝑥 − 3)] Putting x=8 , Perimeter of projection = 2[8 − 5) + (8 − 3)] =2[3+5] =16 Thus, the perimeter of projection is 16ft.