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MArishka [77]
3 years ago
10

50 points! I understand A. and B. but I would really appreciate help with C.

Mathematics
1 answer:
FromTheMoon [43]3 years ago
7 0

Answer:

51.72\text{ cells per hour}

Step-by-step explanation:

So, the function, P(t), represents the number of cells after t hours.

This means that the derivative, P'(t), represents the instantaneous rate of change (in cells per hour) at a certain point t.

C)

So, we are given that the quadratic curve of the trend is the function:

P(t)=6.10t^2-9.28t+16.43

To find the <em>instanteous</em> rate of growth at t=5 hours, we must first differentiate the function. So, differentiate with respect to t:

\frac{d}{dt}[P(t)]=\frac{d}{dt}[6.10t^2-9.28t+16.43]

Expand:

P'(t)=\frac{d}{dt}[6.10t^2]+\frac{d}{dt}[-9.28t]+\frac{d}{dt}[16.43]

Move the constant to the front using the constant multiple rule. The derivative of a constant is 0. So:

P'(t)=6.10\frac{d}{dt}[t^2]-9.28\frac{d}{dt}[t]

Differentiate. Use the power rule:

P'(t)=6.10(2t)-9.28(1)

Simplify:

P'(t)=12.20t-9.28

So, to find the instantaneous rate of growth at t=5, substitute 5 into our differentiated function:

P'(5)=12.20(5)-9.28

Multiply:

P'(5)=61-9.28

Subtract:

P'(5)=51.72

This tells us that at <em>exactly</em> t=5, the rate of growth is 51.72 cells per hour.

And we're done!

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