High School

The population of a slowly growing bacterial colony after \( t \) hours is given by \( p(t) = 2t^2 + 24t + 130 \).

(a) Interpret the meaning of \( p(2) = 186 \).

A. After 186 hours, the colony is growing at an instantaneous rate of 2 bacteria per hour.
B. After 2 hours, the colony is growing at an instantaneous rate of 186 bacteria per hour.
C. After 186 hours, the colony has 2 bacteria in it.
D. After 2 hours, the colony has 186 bacteria in it.
E. None of the above are correct interpretations.

(b) Find the growth rate of \( p(t) \) after 4 hours with correct units.

Answer :

Hence, the growth rate of p(t) after 4 hours is 40 bacteria per hour.

(a) The interpretation of p(2) = 186 is the population of the slowly growing bacterial colony after 2 hours is 186. Therefore, the correct interpretation is "After 2 hours, the colony has 186 bacteria in it."

b) Given that p(t) = 2t² + 24t + 130, the growth rate of p(t) after 4 hours is obtained by calculating p′(4).

Thus, p′(t) = d p(t) / dt = 4t + 24.

Substitute t = 4 in the above equation:

p′(4) = 4(4) + 24

= 16 + 24

= 40.
The growth rate of p(t) after 4 hours is 40 bacteria per hour.

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