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Chapter 2 p D 21 The price is $21 per pizza if 200 large pizzas are being sold each week.
Apply It 2.1 1. a. The formula for the area of a circle is r2 ; where r is the radius. a.r/ D r2
c. To double the number of large pizzas sold, use q D 400. 400 p D 26 40 p D 26 10 p D 16 To sell 400 large pizzas each week, the price should be $16 per pizza.
b. The domain of a.r/ is all real numbers. c. Since a radius cannot be negative or zero, the domain for the function, in context, is r > 0. 2. a. The formula relating distance, time, and speed is d D rt where d is the distance, r is the speed, and t is the time. This can also be written as d 300 t D : When d D 300, we have t.r/ D . r r
4. Revenue D price quantity D pq From the table, the weekly revenue is: pq D 500 11 D 5500 pq D 600 14 D 8400 pq D 700 17 D 11;900 pq D 800 20 D 16;000
b. The domain of t.r/ is all real numbers except 0. c. Since speed is not negative, the domain for the function, in context, is r > 0. 300 d. Replacing r by x: t.x/ D . x 300 600 x x D x D . Replacing r by : t 2 2 x 2 Replacing r by x x 300 1200 Wt D x D . 4 4 x 4
Problems 2.1 1. The functions are not equal because f.x/ 0 for all values of x, while p g.x/ can be p less than 0. For example, f. 2/ D . 2/2 D 4 D 2 and g. 2/ D 2; thus f. 2/ ¤ g. 2/:
e. When the speed is reduced (divided) by a constant, the time is scaled (multiplied) by the r 300c same constant; t D . c r
2. The domain of G is implicitly Œ 3; 1/ while that of H is is implicitly . 1; 1/, so G ¤ H. (We note, however, that for all x in the domain of G; G.x/ D H.x/:/
3. a. If the price is $18.50 per large pizza, p D 18:5. q 18:5 D 26 40 q 7:5 D 40 300 D q At a price of $18.50 per large pizza, 300 pizzas are sold each week.
3. The functions are not equal because they have different domains. h.x/ is defined for all nonzero real numbers, while k.x/ is defined for all real numbers. 4. The functions are equal. For x D 3 we have f.3/ D 2 and g.3/ D 3 1 D 2, hence f.3/ D g.3/. For x ¤ 3, we have x2
4x C 3 .x 3/.x 1/ D D x 1: x 3 x 3 Note that we can cancel the x 3 because we are assuming x ¤ 3 and so x 3 ¤ 0. Thus for x ¤ 3, f.x/ D x 1 D g.x/.
b. If 200 large pizzas are being sold each week, q D 200. 200 p D 26 40 p D 26 5
f.x/ D
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