Question

the number of cells in a tumor doubles every 4.5 months. if the tumor begins with a single cell, how many cells will there be after 3 years? after 6 years?

how many cells will there be after 3 years?
(do not round until the final answer. then round to the nearest whole number as needed.)

how many cells will there be after 6 years?
(do not round until the final answer. then round to the nearest whole number as needed.)

Explanation:

Define the exponential growth model

Using the Exponential Growth and Doubling Time knowledge points, we model the number of cells \(N(t)\) after \(t\) months. Since the initial number of cells is \(N_0 = 1\) and the doubling time is \(d = 4.5\) months, the growth equation is:

$$N(t) = N_0 \cdot 2^{t / d} = 1 \cdot 2^{t / 4.5}$$

Calculate the number of cells after 3 years

First, convert 3 years into months:

$$t_1 = 3 \text{ years} \times 12 \text{ months/year} = 36 \text{ months}$$

Now, substitute \(t_1 = 36\) into the model:

$$N(36) = 2^{36 / 4.5} = 2^8 = 256$$

Calculate the number of cells after 6 years

Convert 6 years into months:

$$t_2 = 6 \text{ years} \times 12 \text{ months/year} = 72 \text{ months}$$

Substitute \(t_2 = 72\) into the model:

$$N(72) = 2^{72 / 4.5} = 2^{16} = 65536$$

Answer:

Question 1

256

Question 2

65536