With data from the Social Security Trustees Report for...

With data from the Social Security Trustees Report for selected years from 1950 and projected to 2030, the number of Social Security beneficiaries (in millions) can be modeled by B(t) = 0.00024t^3 - 0.026t^2 + 1.6t + 2.2 where t is the number of years past 1950. Use the model to find the average number of Social Security beneficiaries per year (actual and predicted) between the following years. (Round your answers to three decimal places.) (a) 1984 and 2002 million per year (b) 2011 and 2021 million per year

Transcript

00:01 Hi there, so for this problem we are given the following f function for this b of d that is equal to 0 .000 24 times the time to the 3 and then this minus 0 .026 times the time square plus 1 .6 times the time 2 .6 times the time plus 2 .2.

00:33 Where the time number is in years, okay? and this passed 1950.

00:39 So the time equals to zero correspond to the year 1950.

00:46 So with that said, we need to use this model to find the average number of social security beneficiaries per year.

00:53 Then for part a between 1984 to 2003.

01:03 So that will be.

01:05 An initial time that is at 34 and a final time after 53 years, okay? because that is the difference between that year and the year 1950, okay? so we know that the average of this function is just going to be the 1 divided by b minus a, the integral from a to b of the function b of d, and then this integrated it over the time.

01:43 Now, we just need to consider the values given for this, so that will be then 1 divided by b, which is 53.

01:55 So we will have that that is.

01:57 Then we have 53 minus 34.

02:07 So that will be 19, and then we do the integral that goes, from a, which is 34 to 53, the function b of d, the one that we are given for this.

02:22 Let me just move it down to here.

02:30 Then we need to integrate this, okay? so the solution of this integral is quite easy, so that will be then.

02:41 Okay, first we will have the solution of the first term, the integral will the first terms, that will be 0 .000224 divided by 3.

02:53 It's already divided by 4.

02:56 Okay, so that will give us 0 .0000006 times the time to the 4 minus, then we will have 0 .026 divided by 3.

03:16 Okay, well we're gonna leave it just like that.

03:19 0 .026 divided by 3 times the time to the 3 plus then we will have 1 .6 divided by 2.

03:32 So that will give us 0 .8 times the time square...

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