311 2.8 Function Operations and Composition 107. Oil Leak An oil well off the Gulf Coast is leaking, with the leak spreading oil over the water’s surface as a circle. At any time t, in minutes, after the beginning of the leak, the radius of the circular oil slick on the surface is r1t2 = 4t feet. Let 1r2 = pr2 represent the area of a circle of radius r. (a) Find 1 ∘ r21t2. (b) Interpret 1 ∘ r21t2. (c) What is the area of the oil slick after 3 min? r(t) 108. Emission of Pollutants When a thermal inversion layer is over a city (as happens in Los Angeles), pollutants cannot rise vertically but are trapped below the layer and must disperse horizontally. Assume that a factory smokestack begins emitting a pollutant at 8 a.m. Assume that the pollutant disperses horizontally over a circular area. If t represents the time, in hours, since the factory began emitting pollutants (t = 0 represents 8 a.m.), assume that the radius of the circle of pollutants at time t is r1t2 = 2t miles. Let 1r2 = pr2 represent the area of a circle of radius r. (a) Find 1 ∘ r21t2. (b) Interpret 1 ∘ r21t2. (c) What is the area of the circular region covered by the layer at noon? 109. (Modeling) Product of Two Factors Suppose that x represents a real number between 0 and 100. Consider two factors of the following forms: N1x2 represents the number that is x less than 100 and G1x2 represents the number that is 20 greater than the product of 5 and x. (a) Express the factor N1x2 in terms of x. (b) Express the factor G1x2 in terms of x. (c) Express their product C1x2. (d) Suppose that in an application C1x2 represents a cost in dollars and the value of x is 20. Evaluate C1202. 110. Software Author Royalties A software author invests his royalties in two accounts for 1 yr. (a) The first account pays 2% simple interest. If he invests x dollars in this account, write an expression for y1 in terms of x, where y1 represents the amount of interest earned. (b) He invests in a second account $500 more than he invested in the first account. This second account pays 1.5% simple interest. Write an expression for y2, where y2 represents the amount of interest earned. (c) What does y1 + y2 represent? (d) How much interest will he receive if $250 is invested in the first account? 111. Sale Pricing In the sale room at a clothing store, every item is on sale for half the original price, plus 1 dollar. (a) Write a function g that finds half of x. (b) Write a function ƒ that adds 1 to x. (c) Write and simplify the function 1ƒ∘ g21x2. (d) Use the function from part (c) to find the sale price of a shirt at this store that has original price $60. 112. Sale Pricing A store is offering 25% off a purchase. A gift card takes an additional $20 off. (a) Write a function g1x2 that finds 25% of x. (b) Write a function ƒ1x2 that adds 20 to x. (c) Write and simplify the function 1ƒ∘ g21x2. (d) Use the function from part (c) to find the amount saved on a $100 purchase.
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