Modeling #7 - Questions and Answers

Question 1 The function C(x) = 17.5x - 10 represents the cost (in dollars) of buying x tickets to the orchestra with a $10 coupon. a. How much does it cost to buy five tickets? b. How many tickets can you buy with $130? Answer 1 a) Here we will put x = 5 into the given function. C(5) = 17.5(5) - 10 C(5) = 77.5 Answer: a) Cost of 5 tickets = $77.5 Answer 2 Step 1: a. In the function c(x) = 17.5x - 10, we substitute x = 5 to obtain the price of purchasing five tickets. C(5) = 17.5 * 5 - 10 Simplifying the expression: C(5) = 87.5 - 10 Therefore, it costs 77.50 to buy five tickets. Step 2: b. To determine the number of tickets that can be purchased with 130, we need to solve the equation C(x) = 130 for x. 17.5x - 10 = 130 Adding 10 to both sides: 17.5x = 140 Dividing both sides by 17.5: x = 140 / 17.5 Hence, with 130, you can buy 140/17.5 tickets, which simplifies to 8 tickets. Answer 3 Solution: a. To find the cost of buying five tickets, we substitute x = 5 into the function C(x) = 17.5x - 10. C(5) = 17.5(5) - 10 Simplifying the expression, we get: C(5) = 87.5 - 10 C(5) = 77.5 Therefore, the cost of buying five tickets is 77.5. b. To find the number of tickets that can be bought with 130, we need to solve the equation C(x) = 130 for x. 17.5x - 10 = 130 Adding 10 to both sides, we get: 17.5x = 140 Dividing both sides by 17.5, we obtain: x = 140 / 17.5 Simplifying the expression, we find: x = 8 Therefore, you can buy 8 tickets with 130. Question 2 A dolphin jumps out of the water, as shown in the diagram. The function h = -16t² + 26t models the height h (in feet) of the dolphin after t seconds. Time to reach 5 feet height. Answer Calculation: According to the question h = -16t² + 26t Also, h = 5 Therefore, 5 = -16t² + 26t Converting the equation in standard form: 16t² - 26t + 5 = 0 On comparing the equation with at² + bt + c a = 16, b = -26, c = 5 Finding the value of t, t = (-b ± √b² - 4ac) / 2a Substituting the values, t = (-(-26) ± √((-26)² - 4(16)(5))) / (2(16)) Opening brackets, t = (26 ± √(676 - 320)) / 32 t = (26 ± √356) / 32 So, t = (26 ± 18.86) / 32 Solving root t = (26 + 18.86) / 32 and t = (26 - 18.86) / 32 So, t = 44.86 / 32 and t = 7.14 / 32 So, t ≈ 1.40 and t ≈ 0.22 Therefore, t ≈ 1.40 and t = 0.22 Time can never be negative, so, time to reach 5 feet height is approximately 0.22 or 1.4 seconds. Question 3 A bacterial culture quadruples in size every hour. You begin observing the number of bacteria 3 hours after the culture is prepared. The amount x of bacteria x hours after the culture is prepared represented by y = 192(4^(x-3)). When will there be 200.000 bacteria? Answer It follows the equation y = 192(4^(x-3)) Calculation: The 200000 represents the value of y. Plug in y as 200000 then solve it for x. Solve the equation for x. Divide both sides by 192. 200000 / 192 = 4^(x-3) 1041.6667 = 4^(x-3) Now, take log on both sides log (1041.6667) = log (4^(x-3)) Divide both sides by log 4 log(1041.6667) / log(4) = x-3 5.0123 = x - 3 Add both sides 3. 8.0123 = x Conclusion: Approximately after 8 hours, the bacteria culture became 200,000. Question 4 The function represents the height y (in feet) of a grasshopper jumping straight up from the ground t seconds after the start of the jump. After how many seconds is the grasshopper 1 foot oft the ground? Given: Function y = -16t² + 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 = -16t² + 8t 16t² - 8t + 1 = 0 (4t - 1)² = 0 t = 1/4 Thus, required time is 1/4 seconds. Question 5 a) To write: A binomial that represents the height of the projection. Given information: Height of wall is x ft and width of projection is (x - 3) ft. Area is represented by x² - 8x + 15. b) To find: Perimeter of projection when height of the wall is 8ft. Given information: Height of wall is x = 8 ft 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, x² - 8x + 15 = x² - 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 projection = (x - 5) ft. Perimeter of projection = 2[(x - 5) + (x - 3)] Putting x = 8, Perimeter of projection = 2[(8 - 5) + (8 - 3)] = 2[3 + 5] = 16 Thus, the perimeter of the projection is 16 ft.