A graphing calculator is recommended. A bacterial culture starts with 300 bacteria and doubles in size every half hour. (a) How many bacteria are there after 3 hours? (b) How many bacteria are there after $$ t $$ hours? (c) How many bacteria are there after 40 minutes? (Round your answer to the nearest whole number.) (d) Graph the population function.

Question

UpStudy AI · Accepted Answer

We are given that the bacterial culture starts with 300 bacteria and that the population doubles every 30 minutes (0.5 hours).

Step 1. Determine the number of doubling periods:
• Every half hour the bacteria double.
• After t hours there are (t / 0.5) = 2t doubling periods.

Now, the general formula for the population after t hours is:
  Population = (initial population) × (2^(number of doubling periods))
  Population = 300 × 2^(2t)

Let’s answer each part.

──────────────────────────────
(a) How many bacteria are there after 3 hours?

For t = 3 hours:
• Number of doubling periods = 2 × 3 = 6.
• Population = 300 × 2^6 = 300 × 64 = 19,200.

So, after 3 hours there are 19,200 bacteria.

──────────────────────────────
(b) How many bacteria are there after t hours?

From the general discussion above, the population after t hours is:
  N(t) = 300 × 2^(2t).

──────────────────────────────
(c) How many bacteria are there after 40 minutes? (Round your answer to the nearest whole number.)

First, convert 40 minutes to hours:
  40 minutes = 40/60 = 2/3 hours.
Now, substitute t = 2/3 into the formula:
  N(2/3) = 300 × 2^(2 × (2/3)) = 300 × 2^(4/3).

To approximate 2^(4/3), note:
  2^(4/3) = e^[(4/3) ln 2]  
  ln 2 ≈ 0.6931, so (4/3) ln 2 ≈ 0.9241.
  Thus, 2^(4/3) ≈ e^0.9241 ≈ 2.52.

Then:
  N(2/3) ≈ 300 × 2.52 ≈ 756.

Rounded to the nearest whole number, there are about 756 bacteria after 40 minutes.

──────────────────────────────
(d) Graph the population function.

The function to graph is:
  N(t) = 300 × 2^(2t).

Key points for graphing (with t in hours) include:
• t = 0: N(0) = 300 × 2^(0) = 300.
• t = 0.5: N(0.5) = 300 × 2^(2×0.5) = 300 × 2^1 = 300 × 2 = 600.
• t = 1: N(1) = 300 × 2^(2×1) = 300 × 2^2 = 300 × 4 = 1,200.
• t = 1.5: N(1.5) = 300 × 2^(2×1.5) = 300 × 2^3 = 300 × 8 = 2,400.
• t = 2: N(2) = 300 × 2^(2×2) = 300 × 2^4 = 300 × 16 = 4,800.
• t = 3: N(3) = 300 × 2^(6) = 300 × 64 = 19,200.

To graph, use the horizontal axis for time t (in hours) and the vertical axis for N(t) (number of bacteria). Plot the points listed above and draw a smooth, continuously increasing exponential curve that passes through them. Label your axes appropriately.

──────────────────────────────
Summary of Answers:
(a) After 3 hours, there are 19,200 bacteria.
(b) After t hours, there are N(t) = 300 × 2^(2t) bacteria.
(c) After 40 minutes, there are approximately 756 bacteria.
(d) The graph of N(t) = 300 × 2^(2t) is an exponentially increasing curve with the key points listed above.
(a) After 3 hours, there are 19,200 bacteria. (b) After t hours, the number of bacteria is $$ N(t) = 300 	imes 2^{2t} $$. (c) After 40 minutes, there are approximately 756 bacteria. (d) The population function $$ N(t) = 300 	imes 2^{2t} $$ is an exponentially increasing curve.

Real Tutor Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Biology Questions