\documentclass[fleqn, 12pt]{article}
%\pagestyle{empty}
\setlength{\topmargin}{-.5in}
\setlength{\oddsidemargin}{-.3in}
\setlength{\textheight}{9in}
\setlength{\textwidth}{7.2in}


\usepackage{amssymb}
\usepackage{amsmath}


\newcommand{\adj}{\mathrm{Adj}}
\newcommand{\mbf}{\mathbf}
\newcommand{\R}{\mathbb{R}}
\newcommand{\ra}{\rightarrow}
\newcommand{\proj}{\mathrm{proj}}
\newcommand{\dotprod}{\bullet}

\newcommand{\sol}{{\textbf{Solution: }}}

\newcommand{\lecture}{{\Large\textbf{Lecture Problems}}}

\begin{document}

\begin{center}
{\Large Warm-Up Problems and Lecture Problems

Solutions

March 19, 2003}
\end{center}

\begin{enumerate}

\item  
\begin{enumerate}
\item  Find the equation of the line connecting the points $(5,10)$ and $(10,0)$.
	\\\sol $y=-2x+20$ or $x=-\frac{1}{2}y+10$.

\item  Consider the region between the $y$-axis and the line segment above.
	Rotate this region about the $y$-axis and draw the resulting solid.
\item  Find the area of a cross section ($A(y)$).
	\\\sol $A(y)=\pi x^2 = \pi(-\frac{1}{2}y+10)^2$.

\item  Find the volume of the solid.
	\\\sol
	\[
	V = \int_0^{10} A(y) \; dy = \int_0^{10} \pi(-\frac{1}{2}y+10)^2 \; dy = \frac{8000\pi}{3}
	\approx 8378
	\]
\end{enumerate}



\item  If $1$ kilogram weighs $2.2$ pounds, how many kilograms is a person who weighs 
	$180$ pounds?
	\\\sol
	\[
	180 \mathrm{lbs} \times \frac{1 \mathrm{kg}}{2.2 \mathrm{lbs}} = 81.8 \mathrm{kg}
	\]

\item   Suppose your dog Fido has a mass of 10 kilograms.
	\begin{enumerate} 
	\item  If Fido steps on your bathroom scale (measuring pounds), what should your scale read?
	\\\sol Do a computation similar to the previous one:
	\[
	10 \mathrm{kg} \times \frac{2.2 \mathrm{lbs}}{1 \mathrm{kg}} = 22 \mathrm{lbs}
	\]
	
	\item  Now you and Fido orbit in the space station with your bathroom scale.
	How much does Fido weigh in orbit?
	(What does your bathroom scale read now?)
	\\\sol Nothing!  Imagine taking a scale into space and stepping on it.  Nothing will register!
	\\
	(I'm trying to illustrate the difference between \emph{mass} and \emph{weight}.)
	\end{enumerate}


	
\end{enumerate}

\end{document}