(need help answering #3 only!!! for the first two problems I included the answers)
The population of bacteria in one cubic centimeter of the blood of a sick person has been modeled by the function P(t)=115t(0.75t) where t is the time, in days, since the person became ill.
Use your calculator to graph the function.
Then use the graph to answer the following questions:
1. To the nearest day, when is the bacteria population at a maximum? Day: 3
2. What is the maximum population? Round your answer to one decimal place. Population: 147.059
3. Estimate how fast the population is changing 14 days after the onset of the illness. Round your answer to two decimal places. Rate of Change:
I have a valid suspicion that P(t)=115t(0.75t)
is supposed to be P(t)=115t(0.75^t), your calculations reflected that.
P'(t) = (115t)(.75^t)(ln.75) + 115(.75^t)
= 0 for a max
(115t)(.75^t)(ln.75) = -115(.75^t)
tln.75 = -1
t = -1/ln.75 = 3.48 , or 3 to the nearest day, you had that, Great!
2) you are correct, I had the same correct to 3 decimals
3) Don't know what your course understands by "estimate".
To me it means: do your work in your head with as little "paperwork" as
possible. Definitely no calculator. I don't know how that would be possible here.
rough work:
(100t)(3/4)^14 (ln.75) + 100(3/4)^t
actual answer: put t = 14 into the derivative
Thank you so much for your help!!!
To estimate how fast the population is changing 14 days after the onset of the illness, we can use the concept of the derivative. The derivative of the function P(t) represents the rate of change of the population with respect to time.
To calculate the derivative of the function P(t), we can differentiate it with respect to t. Here is the step-by-step process:
1. Start with the function P(t) = 115t(0.75t).
2. Take the derivative of each term separately using the power rule and product rule.
dP(t)/dt = 115 * d/dt (t * (0.75t))
3. Apply the product rule by differentiating each term separately.
dP(t)/dt = 115 * [(1 * (0.75t)) + (t * (0.75))] = 115 * (0.75t + 0.75t) = 1.5 * 115t
4. Simplify the expression.
dP(t)/dt = 172.5t
So, the rate of change of the population 14 days after the onset of the illness is 172.5 bacteria per day. Therefore, the estimate for the rate of change is 172.5 (rounded to two decimal places).
To estimate how fast the population is changing 14 days after the onset of the illness, we need to find the derivative of the function P(t). The derivative represents the rate of change of the population with respect to time.
The given function is P(t) = 115t(0.75t). To find the derivative, we can use the power rule and the product rule of differentiation.
1. Apply the power rule to differentiate t(0.75t):
d/dt [t(0.75t)] = 1 * (0.75t) + t * d/dt (0.75t)
= 0.75t + 0.75t
= 1.5t
2. Apply the product rule to differentiate 115t(0.75t):
d/dt [115t(0.75t)] = 115 * d/dt (0.75t) + d/dt [115t * (0.75t)]
= 115 * (1.5t) + (115t) * (1.5)
= 172.5t + 172.5t
= 345t
So, the derivative of P(t) is dP(t)/dt = 345t.
To estimate the rate of change 14 days after the onset of the illness, substitute t = 14 into the derivative:
dP(t)/dt = 345(14)
= 4830.
Therefore, the estimated rate of change 14 days after the onset of the illness is 4830.