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 2 years? after 4 years?

how may cells will there be after 2 years?
40
(do not round until the final answer. then round to the nearest whole number as needed.)

how many cells will there be after 4 years?
(do not round until the final answer. then round to the nearest whole number as needed.)

Explanation:

Formulate the exponential growth model

The number of cells in a tumor grows exponentially. We can model this using the Doubling Time formula:

$$N(t) = N_0 \cdot 2^{\frac{t}{d}}$$

where:

• \(N_0 = 1\) is the initial number of cells.
• \(d = 4.5\text{ months}\) is the doubling time.
• \(t\) is the elapsed time in months.
• \(N(t)\) is the number of cells at time \(t\).

Calculate the number of cells after 2 years

Using the Exponential Growth model, we first convert 2 years into months:

$$t_1 = 2\text{ years} \times 12\text{ months/year} = 24\text{ months}$$

Now, substitute the values into the growth formula:

$$N(24) = 1 \cdot 2^{\frac{24}{4.5}} = 2^{\frac{16}{3}} \approx 40.317$$

Rounding to the nearest whole number gives:

$$N(24) \approx 40$$

Calculate the number of cells after 4 years

Convert 4 years into months:

$$t_2 = 4\text{ years} \times 12\text{ months/year} = 48\text{ months}$$

Substitute the values into the growth formula:

$$N(48) = 1 \cdot 2^{\frac{48}{4.5}} = 2^{\frac{32}{3}} \approx 1625.499$$

Rounding to the nearest whole number gives:

$$N(48) \approx 1625$$

Answer:

Question 1

How may cells will there be after 2 years?
<blank>40</blank>

Question 2

How many cells will there be after 4 years?
<blank>1625</blank>