The graph shows the number of bacteria, y, at time n minutes - HSC - SSCE Mathematics Standard - Question 33 - 2020 - Paper 1

To estimate the value of b, we need to use trial and error:

1. Start with the equation: 2000 = 1000 × b^20.

2. Divide both sides by 1000:

   $2 = b^{20}$

3. Taking the 20th root of both sides gives:

   $b = 2^{1/20}$

4. Utilizing the guess and check method, we can try values for b:

   • For b = 1.03:

     $1000(1.03)^{20} = 1806.11$

   • For b = 1.04:

     $1000(1.04)^{20} = 2191.12$

5. Therefore, we find:

   • Lower estimate: 1.03
   • Upper estimate: 1.04.

Thus, to two decimal places, the estimates for b are:

Lower Estimate: 1.03
Upper Estimate: 1.04.