A cancerous cell lacks normal biological growth regulation a

A cancerous cell lacks normal biological growth regulation and can divide continuously. Suppose a single mouse skin cell is cancerous and its mitotic cell cycle (the time for the cell to divide once) is 20 hours. The number of cells at time t grows according to an exponential model.

(a) Find a formula C(t) for the number of cancerous skin cells after t hours.

(b) Assume a typical mouse skin cell is spherical of radius 50×10^4 cm. Find the combined volume of all cancerous skin cells after t hours. When will the volume of cancerous cells be 1 cm^3?

Solution

a) 1 cell divides into 2 in 20 hrs

Exponential model : C(t) = 1*e^(kt)

  1 cell divides into 2 in 20 hrs

2 = e^(20k)

taking natural log of both sides:

ln2 = 20k ---k = ln2/20 = 0.035

C(t) = e^(0.035t)

b) Volume of cell = (4/3)pi*r^3 = (4/3)*pi*(50 x 10^-4)^3

=5.235 x10^-7 e^(0.035t)

When will the volume of cancerous cells be 1 cm^3

1 =5.235 x10^-7 e^(0.035t)

On solving the above equation by taking natural log on both sides we get:

14.46 = 0.035t

t = 413.22 hours