Week6: Problem 1 Previous Problem (1 point) Suppose that the density of cars (in cars per mile) down a 20mile stretch of the Pennsylvania Turnpike is approximated by 6(x) = 325 (2 + sin (4Vx + 0123)) , at a distance x miles from the Breezewood toll plaza. Sketch a graph of this function for 0 S x S 20. (a) Complete the Riemann sum that approximates the total number of cars on this 20-mile stretch (use Dx instead of Ax): Number = 2 (b) Find the total number of cars on the 20-mile stretch. Number = Week6: Problem 2 Previous Problem Problem List Next Problem (1 point) Find the total mass of the triangular region shown below. All lengths are in centimeters, and the density of the region is 8(x) = 2 + x grams/cm2. -1 1 mass = (include units)Week6: Problem 3 (1 point) The density of oil in a circular oil slick on the surface of the ocean at a distance of r meters from the center of the slick is given by 25 6(r) = 1+r 2 kilograms per square meter. Find the exact value of the mass of the oil slick if the slick extends from r = 0 to r = 15 meters. Mass = Include units in your answer. Week6: Problem 4 (1 point) Water leaks out of a tank through a square hole with 2-inch sides. At time t (in seconds) the velocity of water flowing through the hole is v ft/sec (where v is a function of time, t). Write a definite integral that represents the total amount of water (in cubic feet) lost in the first minute. b Total loss _ fa where a = andb= Week6: Problem 5 (1 point) The dairy industry is an example of cartel pricing: the government has set milk prices artificially high. Sketch a graph of a supply curve (call it s) and a demand curve (call it d). Label the equilibrium price 11* and quantity q*. Then label a price 11+ that is greater than p*. Let q+ be the quantity for which the demand curve passes through p+. Because it's hard to enter p* , p+, etc., in the homework system, let's call the equilibrium price pstar, the equilibrium quantity qstar, and similarly call the increased price pplus and corresponding quantity qplus. (a) On your graph, mark an area corresponding to the consumer surplus when the price is pplus. Express this area as an integral (which may involve some of s, d , and the prices and quantities pstar, pplus. qstar and qplus): b consumer surplus = fa dq, where a = and b = (b) Similarly, on your graph mark an area corresponding to the producer surplus when the price is pplus. Express this area as an integral (which may also involve some of s, d, and the prices and quantities pstar, pplus, qstar and qplus): producer surplus = fab dq, where a = and b = (c) The total gains from trade are given by the sum (Consumer surplus + Producer surplus). How does this sum change when the price is changed from 12* to 12+? ? v Week6: Problem 6 (1 point) If w'(t) is the rate of growth of a child in pounds per year, what does [510 w' (t) (11' represent? BA. The increase in the child's weight (in pounds) between the ages of 5 and 10. C] B. The rate of growth of the child (in lbs/year) between the ages of 5 and 10. C] o. The increase in the child's weight (in pounds) during his/her first five years of life. C] D. The rate of growth of the child (in lbs/year) during his/her first five years of life. Week6: Problem 7 (1 point) A honeybee population starts with 100 bees and increases at a rate of n' (t) bees per week. What does 100 + [015 n' (t) dt represent? A. The total increase in the bee population after 15 weeks. B. The total bee population after 15 weeks. C. The average rate at which the bee population increases over the rst 15 weeks. D. The rate at which the bee population increases after 15 weeks have passed. Week6: Problem 8 (1 point) The voltage, V, in an electrical outlet is given as a function of time, t, by the function V = Tcos (at), where V is in volts and t is in seconds, T is a positive constant representing the maximum voltage, and a is a positive constant related to the frequency of the delivered voltage. (a) If a = 1001:, what is the average value of the voltage over 1 second? average = (b) Engineers do not use the average voltage. They use the root mean square voltage defined by V = \\/ average of (V2). Find V in terms 0f T. (Take the average over 1 second.) 17 = (e) The standard household voltage in parts of Japan is 100 volts, meaning that 17 = 100. What is T? T = an America, the household voltage is 110 volts, at a frequency of 60 Hz, so that a = 1207:.) Week6: Problem 9 (2 points) Coal gas is produced at a gasworks. Pollutants in the gas are removed by scrubbers, which become less and less efficient as time goes on. The following measurements, made at the start of each month, show the rate at which pollutants are escaping (in tons/month) in the gas: 012 3 561016 4 23 Time (months) Rate 5 5 3142 A. Make an overestimate and an underestimate of the total quantity of pollutants that escape during the first month. overestimate = tons underestimate = tons B. Make an overestimate and an underestimate of the total quantity of pollutants that escape for the whole six months for which we have data. overestimate = underestimate =