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May 2001: Paper 2 #5

The number ( n ) of bacteria in a colony after h hours is given by the formula . Initially, there are 1200 bacteria in the colony. May 2001: Paper 2 #5.

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May 2001: Paper 2 #5

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  1. The number (n) of bacteria in a colony after h hours is given by the formula . Initially, there are 1200 bacteria in the colony. May 2001: Paper 2 #5 (a) Copy and complete the table below, which gives values of n and h. Give your answers to the nearest hundred. 1600 3600 Be sure to copy the entire table in your answer booklet! When h = 1, we have When h = 4, we have

  2. 4000 Number of bacteria (n) (3,2700) (4,3600) 3000 (1,1600) 2000 (2,2100) 1000 (0,1200) 1 2 3 4 Time in hours (h) (b) On graph paper, draw the graph of the above function. Use a scale of 3 cm to represent 1 hour on the horizontal axis and 4 cm to represent 1000 bacteria on the vertical axis. Label the graph clearly.

  3. (c) Use your graph to answer each of the following showing your method clearly. • How many bacteria would there be after 2 hours and 40 minutes? Give your answer to the nearest hundred. • After how long will there be approximately 3000 bacteria? Give your answer to the nearest 10 minutes. The key to this question is to using the graph that you have already made. You need to use broken lines on the graph that clearly show the ordered pairs. For part (i), you do this by finding 2 hours 40 minutes on the horizontal axis and showing which value on the vertical axis corresponds to this time. For part (ii), you start on the horizontal axis and find 3000 bacteria. You are then to show which value on the horizontal axis corresponds to this amount.

  4. 3000 bacteria 2500 2 hrs 40 min 3 hrs 20 min 4000 At 2 hrs 40 min, there are about 2500 bacteria. Number of bacteria (n) 3000 2000 There will be 3000 bacteria at 3 hrs 20 min. 1000 1 2 3 4 Time in hours (h)

  5. 5cm x x (ii) A picture is in the shape of a square of side 5 cm. It is surrounded by a wooden frame width x cm, as shown in the diagram below. l The length of the wooden frame is l cm, and the area of the wooden frame is A cm2. (a) Write an expression for the length of l in terms of x. 2x + 5. l = x + 5 + x =

  6. 5cm x x (b) Write an expression for the area A in terms of x. The area we are looking for is only the area of the frame. The area of the picture is given by (5 cm)(5 cm) = 25 cm2. The total area inside the frame is given by A = (total area inside frame) – (area of picture) Substituting these values, we see l

  7. Note: You could just as easily used the quadratic formula to solve… (c) If the area of the frame is 24 cm2, find the value of x. From part (b), we know Substituting A = 24, we have Subtracting 24 from both sides gives us Factoring completely yields Thus, by the zero product property, we know x = –6 or x = 1 Since x is a length, x –6. Therefore, x = 1.

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