Despite efforts at cost containment, the cost of the...

Despite efforts at cost containment, the cost of the Medicare program is increasing. Two major reasons for this increase are an aging population and extensive use by physicians of new technologies. Based on data from the Health Care Financing Administration and the U.S. Census Bureau, health-care spending through the year 2000 may be approximated by the function $$ \begin{array}{l} S(t)=0.02836 t^{3}-0.05167 t^{2}+9.60881 t+41.9 \\ \quad(0 \leq t \leq 35) \end{array} $$ where $S(t)$ is the spending in billions of dollars and $t$ is measured in years, with $t=0$ corresponding to the beginning of 1965 . a. Find an expression for the rate of change of health-care spending at any time $t$. b. How fast was health-care spending changing at the beginning of $1980 ?$ At the beginning of $2000 ?$ c. What was the amount of health-care spending at the beginning of $1980 ?$ At the beginning of 2000 ?

Transcript

0:00 Hi there.

00:01 So for this problem, we are told that based on the data from the healthcare financing administration, the health care spending through the year 2000 may be approximated by the following expression, that is, that adds of t is equal to 0 .0 ,2 ,836.

00:31 The time to the 3 and then this minus 0 .0 5 ,167 times the time square.

00:42 And then we have another term in here that is 9 .681 times the time and then this plus 41 .41 times the time.

00:56 And then this plus 41 .49.

01:03 This between the times equals to 0 and 35.

01:11 So the question, well, remember that for this expression, the times zero corresponds to the year 1965.

01:23 So for part a of this problem, we are asked to find an expression for the rate of change of the health care expanded at any time.

01:32 So that will be just simply the derivative of the expression that we are given with respect to time.

01:38 So that will be 3 times 0 .0, 2 ,836.

01:46 And this times the time is square.

01:49 This minus 2 times 0 .0 5 ,167 times the time.

01:59 And this plus 9 .60881.

02:07 So we can simplify this border and this is equal to.

02:18 Okay, so let me calculate for the first turn in here.

02:27 So that will give you a value of 0 .0 ,8 ,508 times the time is squared.

02:45 This minus 2 times.

02:50 0 .05 ,167.

02:55 This will give us 0 .10 ,334 times the time plus 9 .60 ,881.

03:09 So that's the solution for part a of this problem.

03:14 Now for the next question in here is about how fast was health care expanding changing at the beginning of 1980.

03:25 So we know that this corresponds to the time 15 years of um yes 15 years after 1965.

03:36 So what we need to do is to simply evaluate the function that we obtain from part a at 15.

03:42 So that will be 0 .08 ,008 times 15 to the 15 to the square and this minus 0 .10 ,334 times 15, this plus 9 .60 ,881.

04:11 Then using our calculator, we obtain a value of so the value that we equal to 27 point well let me see well i'm going to include two in two a decimal places well no sorry i'm going to include all of the decimals that i obtained from this that will be 27 .271 so that's the solution for part of part b of this problem now for p c the question is what was the amount of the health at the beginning.

05:08 Oh, sorry, no.

05:09 It was the next one is at the beginning of 2000.

05:16 So the time for this is 35.

05:19 So we just need to evaluate the function that we obtained from part a at 35.

05:25 Then that will be 0 .0 ,8 ,508 times 35 to this square.

05:39 Minus 0 .10 ,334 times 35...

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