The concentration C, in parts per million, of a medication...

The concentration $C,$ in parts per million, of a medication in the body $t$ hours after ingestion is given by the function $C(t)=10 t^{2} e^{-t}$ a) Find the concentration after $0 \mathrm{hr}, 1 \mathrm{hr}, 2 \mathrm{hr}, 3 \mathrm{hr},$ and $10 \mathrm{hr}$. b) Sketch a graph of the function for $0 \leq t \leq 10$. c) Find the rate of change of the concentration, $C^{\prime}(t)$. d) Find the maximum value of the concentration and the time at which it occurs. e) Interpret the meaning of the derivative.

Transcript

00:01 Hello everyone, in this question it is given that c of t equal to 10 t square e to the power minus t.

00:08 So the first part of the question is we have to find for each point that is c of 0 will be 10 0 square e to the power minus 0 which will be 0 paths per million.

00:20 Next c of 1 c of 1 will be 10 1 square into e to the power minus 1 that will be 20 by e paths per million.

00:32 Then c of 2 c of 2 will be 10 into 2 square into e to the power minus 4 that will be 40 by e square paths per million.

00:44 Then c of 3 equal to 10 into 3 square into e to the power minus 2 that will be 90 divided by e to the power 3 paths per million.

00:58 Then c of 10 will be 10 into 10 square e to the power minus 10 that will be 1000 divided by e to the power 10 paths per million.

01:16 So this is the first part of the question.

01:18 The second part of the question is they have to sketch the graph.

01:22 So the graph will look like this.

01:24 This is the origin and this is c of t and this is t.

01:30 So we will be adding like this.

01:34 So at t equal to 2 and varies from 10.

01:40 It will go like it will end with here.

01:46 Then the third part is c dash of t c dash of t is d by dt 10 t square e to the power minus t.

01:57 So we will be having 10 d by dt t square e to the power minus t.

02:05 So this will lead to 10 t square minus e to the power minus t then plus e to the power minus t 2t.

02:19 So if you solve this e to the power minus t 10 t square minus 10 t square plus 20t.

02:28 Then the fourth part is to find out the maximum value.

02:34 So for maximum value equate c dash of t equal to 0 that is e to the power minus t minus 10 t square plus 20t equal to 0...

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