The population of a slowly growing bacterial colony after t hours is given by p(t)=3t2+29t+150 . Find the growth rate after 4 hours?
Like Drwls said above it's really a differential Calc question. Just take the differative of p(t)
p'(t) = 6t + 29 + 0 .
p'(4) = 6(4) + 29 = 53 bacteria/hr
Well, if we want to find the growth rate, we need to find the derivative of the population function. So let me put on my math hat and do some calculations.
First, let's find the derivative of p(t):
p'(t) = d/dt(3t^2 + 29t + 150)
= 6t + 29
Now, we can find the growth rate after 4 hours by plugging in t = 4 into the derivative:
p'(4) = 6(4) + 29
= 24 + 29
= 53
So, the growth rate after 4 hours is 53. However, I must say, that's a mighty impressive growth rate for a bacterial colony! Those bacteria must have been doing some serious calisthenics. Keep an eye out for any tiny dumbbells and sweatbands the next time you observe the colony.
To find the growth rate after 4 hours, we need to find the derivative of the population function p(t) with respect to t and evaluate it at t = 4.
The population function is given by p(t) = 3t^2 + 29t + 150.
To find the derivative, we can use the power rule for differentiation. For a term of the form at^n, the derivative is given by n * a * t^(n-1).
Taking the derivative of p(t) term by term, we have:
dp/dt = d/dt (3t^2) + d/dt (29t) + d/dt (150)
= 6t + 29 + 0
= 6t + 29.
Now, we can evaluate the derivative at t = 4:
dp/dt | (t=4) = 6(4) + 29
= 24 + 29
= 53.
Therefore, the growth rate after 4 hours is 53.
To find the growth rate of the bacterial colony after 4 hours, we need to find the derivative of the population function with respect to time (t). The derivative will give us the rate of change of the population with respect to time.
Given the population function:
p(t) = 3t^2 + 29t + 150
To find the derivative, we apply the power rule of differentiation. For each term in the function, we multiply the coefficient of the term by the exponent and then subtract 1 from the exponent. The derivative of the population function gives us the growth rate of the colony.
Taking the derivative of the given population function:
p'(t) = d/dt (3t^2 + 29t + 150)
= 6t + 29
The derivative of the population function with respect to time gives us the growth rate of the bacterial colony.
Substituting t = 4 into the derivative expression gives us the growth rate after 4 hours:
p'(4) = 6(4) + 29
= 24 + 29
= 53
Therefore, after 4 hours, the growth rate of the bacterial colony is 53.
The growth rate is the derivative of the population, dp/dt.
dp/dt = 6t + 29
when t = 4, dp/dt = 24 + 29 = 53 bacteria per hour.
This exact method is based upon using differntial calculus. If you are unfamiliar with it, try calculating p at 4.05 and 3.95 h, taking the difference, and dividing by the time interval, 0.1 hours.
p(4.05) = 316.66
p(3.95) = 311.36
change during interval = 5.30
rate of change = 53