Mathematics 1228A/B Study Guide - Midterm Guide: Black Cards, Ath, Bernoulli Trial

Mark and Nancy are making punch for a party and are using two types of juice: pineapple and papaya. The total number of quarts must be more than 9 but less than 15. They should use at least 4 quarts of each type. We want to find out how many quarts of each juice they should buy. First graph the system below and describe the shape of the solution area. Choose one answer. rectangle circle triangle square Mark and Nancy are making punch for a party and are using two types of juice: pineapple and papaya. The total number of quarts must be more than 9 but less than 15. They should use at least 4 quarts of each type. We want to find out how many quarts of each juice they should buy. List any two ordered pair solutions from your graph in #1. Choose one answer. (3, 4), (6, 10) (-1, 9), (3, 7) (8, 8), (2, 12) (5, 6), (4, 7) Mark and Nancy are making punch for a party and are using two types of juice: pineapple and papaya. The total number of quarts must be more than 9 but less than 15. They should use at least 4 quarts of each type. We want to find out how many quarts of each juice they should buy. Pick one of the ordered pair solution from your answer in #2 and explain what each number represents. Choose one answer. the x-value represents the number of papayas and the y-value represents the number of pineapples the x-value represents the y-axis and the y-value represents the x-axis the x-value represents the number of quarts of pineapple and the y-value represents the number of quarts of papaya juice the x-value represents the number of people drinking pineapple juice and the y-represents the number of people not drinking pineapple juice

Step 3 Multiply both sides of the differential equation by t and get [ty] = dt Step 4 Integrating both sides and solving for y, we obtain C1C- 2 Int C To satisfy the initial condition, we must have 21n(1)+C-C [In(1)-이. Hence, the solution of the initial-value problem is 2Int 1 Figure 3 Solution of the initial-value problem fy + ty 2, y(1)= 1, t > 0 t > 0 Among all the solution curves that are shown in Fig. 3, th problem is the curve that goes through the point (1, 1) e solution of the > Now Try Exercise In the following section, we present several interesting applications of fre linear differential equations. Check Your Understanding 10.3 1. Using an integrating factor, solve yty 1e 2. Find an integrating factor for the differential equi EXERCISES 10.3 in Exercises 1-6, find an integrating factor for each equation Taket0. In Exercises 21-28, solve the initial-value problem. 22. ty, + y = In t, y(e) = 0, t > 0 Int-1 23. y+ 24, y, 2(10-y), y(0) = 1 10-9e-2t 10+10f In Exercises 7-20, solve the given equation using an integrating factor. Taket>0. 26. ty'-y=-1, y(1) = 1, t>01 27, y, + 2y cos(2t)-2 cos(2t), y(5) = 0 y 28, ty' + y-sin t, y()-0, t > 0-1 cost Technology Exercises 29. Consider the initial-value -1- - 9. 10· y' = 2(20-y) 20 + C'e-2t 12. e +1 11、 y' = .5(35-y) 13. y, + 10+1-0y=府14.3-h-e2t tr" + Cee 15. (1 + tly, + y--I 17. 6y'+ty t 19,y, + y = 2-et y'=-th+10, (a) Is the solution increasing or decreasing whie y(0)=50. 16, y, = e-"(y + 1) _1+ 18. ety' + y = 1 1 + Ce", C'e-e-t 20 , + y = 1 [Hint: Compute y'(0),1 i + Ce-i(t+1)3/2 (b) Find the solution and plot it for OStS t k indicates answers that are in the back of the book. 19. y = 2-ne, + Ce-r 29. (a) y'(0)--40, y(t) is decreasing at t = 0