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Question

# The monthly Medicare Part B healthcare premium for most beneficiaries ages $65$ and older has increased significantly since 1975. The monthly premium has increased from about $\ 7$ in 1975 to $\ 110.50$ in 2011. The following exponential function models the premium increases:$M(x)=7(1.080)^x,$where $x$ is the number of years since 1975. Estimate the monthly Medicare Part B premium in 1985, 1992, and 2002. Round to the nearest dollar.

Solution

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Answered 1 year ago
Answered 1 year ago
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By substituting in the function

\begin{aligned} M\left(x\right)=7\cdot\left(1.080\right)^x \end{aligned}

, for the values $x=10,17$, and $x=27$, respectively, where $x$ represents the number of years since $1975$, the following is true:

\begin{aligned} M\left(\textcolor{#4257b2}{10}\right)&=7\cdot\left(1.080\right)^{\textcolor{#4257b2}{10}}\\ &=7\cdot2.158925\\ &=15.112475\\ &\approx\15\\ M\left(\textcolor{#4257b2}{17}\right)&=7\cdot\left(1.080\right)^{\textcolor{#4257b2}{17}}\\ &=7\cdot3.7\\ &=25.9\\ &\approx\26\\ M\left(\textcolor{#4257b2}{27}\right)&=7\cdot\left(1.080\right)^{\textcolor{#4257b2}{27}}\\ &=7\cdot7.988\\ &=55.916\\ &\approx\56 \end{aligned}

Therefore, the monthly Medicare Part B premium in $1985$ was $\15$, in $1992$ was $\26$, and in $2002$ was $\56$.

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