The value of p has three significant positions the

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The value of P has three significant positions. The precision of a - t is harder to estimate, but if the exponent (time) is reasonably accurate, we can make a good guess. Notice that a 0 = 1 is an exact value, and the value a - 27 = 0.86 has just two significant positions. For values of t between 0 and 27, we can expect at least two significant positions in a - t . To avoid loss of precision, we should use either the algebraic value of a or the stored calculator value of a to obtain the best estimate of f ( 13 ) : f ( 13 ) = 1.27 a - 13 1.0526557543. But this answer is way too precise! Based on the data, P has three significant posi- tions and a - 13 has at least two significant positions, so the product should have between two and three significant positions. We will display our answer to three significant posi- tions: f ( 13 ) 1.05 1.05 ppm. Note: If our time had been closer to 27 hours, we would expect our answer to have closer to two significant positions. If our time had been more than 27 hours we would expect a greater loss of precision. 1 parts per million 7
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5 Applying What We Learned Problem 3. In an optimal environment, a bacteria population grows exponentially. At noon in the sewage treatment plant, experimenters introduce specially cultured bacteria into a barrel full of rich nutrients. We will assume that the bacteria population in the barrel is modeled by an exponential function. Let N ( t ) be the number of bacteria after t days. Then N ( t ) = Pa t for some constants P and a . Measurements indicate that N ( 3 ) = 5400 and N ( 9 ) = 374, 000 . a. Before working the problem, estimate (guess!) the value of N ( 6 ) , the number of bac- teria at noon on the fourth day. b. Write down two equations for P and a , one when t = 3 and the other when t = 9 . c. Use these two equations to compute a and P . Round your values to three significant digits, but be sure to store the more precise values for further calculations. d. Write down the formula for N ( t ) using the rounded values of a and P . (But remember to use the more precise stored values when you use the formula!) e. Evaluate N ( 6 ) , N ( 10 ) , and N ( 14 ) . (Give answers to the nearest bacterium.) f. Does your guess in (a) agree with your answer in (e)? g. Is the precision of your answers in part (e) in agreement with the measurements for N ( 3 ) and N ( 9 ) ? h. To what extent should the values in (e) be rounded? Explain your answer. 8
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Math 148 – Lab 3 Name: Answer Sheet Recitation Instructor: Page 1 of 2 Recitation Time: (1) (a) (b) (i) P = (ii) a = (iii) Equation: (c) (i) P = (ii) r = (iii) Equation: (d) (i) using (b): (ii) using (c) f ( 22 ) = f ( 22 ) = (iii) Explanation: (e) a = (2) (a) (i) f ( t ) = (ii) f ( 9.5 ) = (b) (i) [Nothing to write] (ii) f ( 9.5 ) = (c) error = Scratch Pad: Anything below this line will be ignored by the grader.
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