Question

based on data from 1940 to 2000, the average yearly social security benefits for retired individuals in the united states has been modeled by (s(t)=0.348t^{2}-5.5t + 35.11) in thousands of dollars, where (t) is time in years since the start of 1940 ((0leq tleq60)). (site: https://www.ssa.gov/policy/docs/quickfacts/stat_snapshot/) a. calculate the derivative of (s(t)). (s(t)=) b. evaluate (s(20)) and use it to interpret in the context of the problem. (do not round.) the average yearly social security benefits are ? by

Explanation:

Step1: Apply power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y'=nax^{n - 1}$. For $S(t)=0.348t^{2}-5.5t + 35.11$, the derivative of $0.348t^{2}$ is $2\times0.348t^{2 - 1}=0.696t$, the derivative of $-5.5t$ is $-5.5$, and the derivative of the constant $35.11$ is $0$. So $S'(t)=0.696t-5.5$.

Step2: Evaluate $S'(20)$

Substitute $t = 20$ into $S'(t)$. We have $S'(20)=0.696\times20-5.5$.
$S'(20)=13.92 - 5.5=8.42$.
In the context of the problem, since $S(t)$ is in thousands of dollars and $t$ is in years, $S'(20)$ represents the rate of change of the average yearly Social Security Benefits. So the average yearly Social Security Benefits are increasing by $8.42$ thousand dollars per year in the year $1940 + 20=1960$.

Answer:

a. $S'(t)=0.696t - 5.5$
b. The average yearly Social Security Benefits are increasing by $8.42$ thousand dollars per year in 1960.