A bacteria culture initially contains 1500 bacteria and doubles every half hour.

a) Find an expression for the number of bacteria after t hours.
Q(t)=


b) The number of bacteria after 20 minutes is (the answer must be an integer)

c) The number of bacteria after 9 hours is

(a) 1500*2^(t/0.5) = 1500*2^(2t)

(b) 1500*2^(2*20/60) = ?
(c) 1500*2^(2*9) = ?

a) The bacteria doubles every half hour, so we can express the growth as an exponential function. Let's call the number of bacteria at time t as N(t). In t hours, there are 2^[(t / (1/2))]=(2^2t) times the initial number of bacteria.

Therefore, the expression for the number of bacteria after t hours is:

Q(t) = 1500 * 2^(2t)

b) We are given that 20 minutes is equal to 20/60 = 1/3 hours.

To find the number of bacteria after 20 minutes, we substitute t = 1/3 into the expression for Q(t):

Q(1/3) = 1500 * 2^(2 * (1/3))
= 1500 * 2^(2/3)
≈ 1500 * 2^(0.6667)
≈ 1500 * 1.5874
≈ 2381.1

The number of bacteria after 20 minutes is approximately 2381.

c) We are given that 9 hours is equal to 9 * 60 = 540 minutes, and 540 minutes is equal to 540/60 = 9/1 = 9 hours.

To find the number of bacteria after 9 hours, we substitute t = 9 into the expression for Q(t):

Q(9) = 1500 * 2^(2 * 9)
= 1500 * 2^18
≈ 1500 * 262144
≈ 393216000

The number of bacteria after 9 hours is approximately 393,216,000.

To find the expression for the number of bacteria after t hours, we need to determine the growth rate of the bacteria. We know that the bacteria doubles every half hour.

a) The doubling time is 0.5 hours, which means the bacteria's growth rate is 2^(1/0.5) = 2^2 = 4 (since the bacteria doubles every 0.5 hours). The initial number of bacteria is 1500.

Therefore, the expression for the number of bacteria after t hours is given by: Q(t) = 1500 * (4^t).

b) To find the number of bacteria after 20 minutes, we need to convert the time to hours. Since 1 hour = 60 minutes, 20 minutes is equal to 20/60 = 1/3 hours.

Substituting t = 1/3 into the expression Q(t) = 1500 * (4^t), we get: Q(1/3) = 1500 * (4^(1/3)) ≈ 1500 * 1.587401 = 2381.1015.

Since the number of bacteria must be an integer, we round down to the nearest whole number: Q(1/3) ≈ 2381.

Therefore, the number of bacteria after 20 minutes is approximately 2381.

c) To find the number of bacteria after 9 hours, we substitute t = 9 into the expression Q(t) = 1500 * (4^t).

Q(9) = 1500 * (4^9) = 1500 * 262144 = 393216000.

Therefore, the number of bacteria after 9 hours is 393,216,000.