Answer:
• n(t) = 150·2^t . . . . . . . . . . . . number of cells after t minutes
• a(t) = π(0.25 +0.50t)^2 . . . . area in cm^2 after t minutes
• d(t) = n(t)/a(t) = (2400·2^t)/(π(1+2t)^2)
Explanation:
The number of cells (n(t)) is described by an exponential function of time (t) with an initial value of 150 and a growth factor of 2 each minute:
n(t) = 150·2^t . . . . . . n in cells; t in minutes
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The area of the culture is given by ...
a(t) = π·(r(t))^2 . . . . where r(t) is the radius as a function of time.
The radius is linearly increasing with a rate of increase of 0.50 cm/min, so can be described by ...
r(t) = 0.25 +0.50t
Then the area is ...
a(t) = π·(0.25 +0.50t)^2
A factor of 0.25 can be removed from inside parentheses to make this be ...
a(t) = (π/16)(1 +2t)^2 . . . . . a in cm^2; t in minutes
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The density is the number of cells divided by the area:
d(t) = n(t) / a(t) = 150·2^t/((π/16)(1 +2t)^2)
Simplifying a bit, this is ...
d(t) = (2400/π)(2^t)/(1 +2t)^2 . . . . . d in cells/cm^2; t in minutes