4.6 Area Between Curves and Producers' and Consumers' Surplus e EXERCISES Basic SkiLLs PrRACTICE For Exercises 1 - 6, find the area of the region between the curves on the given interval. 1. y=x'+2and y = x2 on the interval [0,2] 2. y==x>+4x-1and y = x+5 on the interval [-1,3] 3. y=2x+1and y = x-8 on the interval [-7,-4] 4. y=-x+1and y = 3x+ 6 on the interval [-3,2] 5. y= X' =5and y=-2x+10 on the interval [1.6] 1 3 6. y==0.5x"+16 and y = 0 on the interval [-2,4] For Exercises 7 - 9, the price-demand equation and equilibrium point for a certain commodity are given. Use the information to find the consumers' surplus for the commodity. 7. p=D(x)=-0.8x+ 150, and the equilibrium point is (25, 130). 8. p=D(x)=-3x+ 1000, and the equilibrium point is (200,400). 9. p=D(x)=-0.5x+200, and the equilibrium point is (75, 162.50). For Exercises 10 - 12, the price-supply equation and equilibrium point for a certain commodity are given. Use the information to find the producers' surplus for the commodity. 10. p = 5(x) = 5.2x, and the equilibrium point is (25, 130). 11. p=S8(x)=0.4x+ 308, and the equilibrium point is (290,424). 12. p=S(x) =0.03x+50, and the equilibrium point is (45,51.35). 13. The price-demand equation for a particular denim jacket is given by p = 206 0.2x dollars, which gives the price per jacket when x jackets are demanded. If market equilibrium is reached when 500 jackets are sold at a price of $106.00 each, find the consumers\" surplus. 14. The price-supply equation for the denim jacket described in the previous example (Example #13) is given by p = 0.04x+ 86 dollars, which gives the price per jacket when x jackets are supplied. Find the producers' surplus. Chapter 4: Antidifferentiation 15. The supplier of a popular board game called Who Dunnit? has a price-supply equation given by p = 0.05x + 10 dollars, which gives the price per game when x games are supplied. If market equilibrium is reached when 300 games are sold at a price of $25 each, find the producers' surplus. 16. The price-demand equation for the board game described in the previous example (Example #15) is given by p = 28 - 0.01x dollars, which gives the price per game when x games are demanded. Find the consumers' surplus. 17. During 2018, a company that sells clothes dryers had a weekly marginal cost function given by C '(x) = 12+ 0.4x dollars per dryer, where x is the number of dryers produced each week. During 2019, the company had a weekly marginal cost function given by C> "(x) = 10+ 0.3x dollars per dryer, where x is the number of dryers produced each week. (a) If 200 dryers were produced each week during both years, during which year did the company have higher weekly costs? (b) How much more did the company pay in weekly costs during that year? 18. Two rival dog treat companies, Daisy-licious and Tetra-tastic, have production rates given by Pp'(H)=50+ %: and Pr'(1) =50+ %f bags of treats per hour, respectively, hours after they start making bags of treats. (a) Aftera 12 hour work shift, which company produces more bags of treats? (b) How many more bags of treats does this company produce than the other during the 12 hour work shift? Round your answer to the nearest integer, if necessary. 19. Super Rad Skateboardz specializes in selling glow in the dark skateboards. During March of 2019, the company had a monthly marginal revenue function given by R, '(x) = 200 - 1.6x dollars per skateboard, where x is the number of skateboards sold. During March of 2020, Super Rad Skateboardz had a monthly marginal revenue function given by R "(x) = 90 2.4x dollars per skateboard, where x is the number of skateboards sold. (a) If the company sold 35 skateboards during March of both 2019 and 2020, during which year did the company have a higher monthly revenue? (b) How much greater was the monthly revenue? 20. Two objects start from the same location and travel along the same path. Object A has velocity v4 (1) = 0.5 +6, and object B has velocity vg(r) = 0.3r + 8, both measured in meters per second, after seconds. How far ahead is object B after 5 seconds? INTERMEDIATE SKILLS PRACTICE For Exercises 21 - 24, find the area of the region between the curves on the given interval. 21. y=x\" and y = 4 on the interval [-1,5] 22. y=6x>-3x2+x-4 and y = x+7 on the interval [-6,-2]