20_Cal_Solution of Calculus_6e - 99 45 = a 207 Â 35 = a 72...

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0.5 1 1.5 2 2.5 3 20 40 60 80 100 120 x ( t ) = 60 t if 0 t 0 . 5 , 60 60 t if 0 . 5 < t 1 , 60 t 60 if 1 < t 3 . The graph of x ( t ) is shown next. C01S02.077: Initially we work in units of pages and cents (to avoid decimals and fractions). The graph of C , as a function of p , must be a straight line segment, and its slope is (by information given) C (79) C (34) 79 34 = 305 170 79 34 = 135 45 = 3 . Thus C ( p ) = 3 p + K for some constant K . So 3 · 34+ K = 170, and it follows that K = 68. So C ( p ) = 3 p +68, 1 p 100, if C is to be expressed in cents. If C is to be expressed in dollars, we have C ( p ) = (0 . 03) p + 0 . 68 , 1 p 100 . The “fixed cost” is incurred regardless of the number of pamphlets printed; it is $0 . 68. The “marginal cost” of printing each additional page of the pamphlet is the coefficient $0 . 03 of p . C01S02.078: We are given C ( x ) = a + bx where a and b are constants; we are also given 99 . 45 = C (207) = a + 207 b and 79 . 15 = C (149) = a + 149 b. Subtraction of the second equation from the first yields 20 . 3 = 58 b , so that b = 0 . 35. Substitution of this
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Unformatted text preview: 99 . 45 = a + 207 · . 35 = a + 72 . 45 , so that a = 27 . Therefore C ( x ) = 27 + (0 . 35) x , 0 5 x < + ∞ . Thus if you drive 175 miles on the third day, the cost for that day will be C (175) = 88 . 25 (in dollars). The slope b = 0 . 35 represents a cost of $0 . 35 per mile. The C-intercept a = 27 represents the daily base cost of renting the car. In civil engineering and in some branches of applied mathematics, the intercept a = 27 is sometimes called the ofset, representing the vertical amount by which C (0) is “o±set” from zero. C01S02.079: Suppose that the letter weighs x ounces, 0 < x 5 16. If x 5 8, then the cost is simply 8 (dollars). If 8 < x 5 9, add $0 . 80; if 9 < x 5 10, add $1 . 60, and so on. Very roughly, one adds $0 . 80 if 9...
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