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{SECT 0 {PARA 259 "" 0 "" {TEXT -1 32 "Height, Weight, and Surface Are
a" }}{PARA 256 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA 257 "" 0 ""
{TEXT -1 12 "Georgia Tech" }}{PARA 258 "" 0 "" {TEXT -1 21 "herod@math
.gatech.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 288 " There is a correlation be
tween the height of a sphere, its volume, and its surface area. We giv
e the volume and surface area of a sphere as a function of how high th
e sphere stands above the floor. Then we compute the rate of change of
volume and surface area with respect to height." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "V:= h -> 4*P
i*h^3/24; \nS:= h -> Pi*h^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 29 "diff(V(h),h); \ndiff(S(h),h);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 158 " We ask about humans. How does t
he weight and surface area of a human change with height. We provide d
ata from height/weight chart for medium built males." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "ht:=[62.,
63.,64.,65.,66.,67.,68.,69.,70.,71.,72.,73.,74.];\nwt:=[128.,131.,135.
,139.,142.,146.,150.,154.,158.,162.,\n 167.,172.,177.];\npts:=[s
eq([ht[i],wt[i]],i=1..13)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
22 "plot(pts,style=POINT);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 " \+
We assume there is a relationship between height and weight in the
form" }}{PARA 0 "" 0 "" {TEXT -1 30 " "
}{XPPEDIT 18 0 "w = k*h^n;" "6#/%\"wG*&%\"kG\"\"\")%\"hG%\"nGF'" }
{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 98 "We wish to find n and k. Here's one way to try to do tha
t: take the logarithm of both sides. Then" }}{PARA 0 "" 0 "" {TEXT -1
38 "\n\011\011\011\011\011 Ln(w) = n Ln(h) + Ln(k)." }}{PARA 0
"" 0 "" {TEXT -1 41 "\nThus, we plot the logarithm of the data." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
44 "lnpts:=[seq([ln(ht[i]),ln(wt[i])],i=1..13)];" }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 24 "plot(lnpts,style=POINT);" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We get a least squares fi
t for this data and plot the result." }}{PARA 0 "" 0 "" {TEXT -1 0 ""
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(stats);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "xcord:=map(ln,ht);\nycord:=map(ln,w
t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "fit[leastsquare[[x,y
],y=a*x+b,\{a,b\}]]([xcord,ycord]);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 12 "ans:=rhs(%);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "op(2,ans);\nA:=exp(op(2,an
s));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k:=op(1,op(1,ans));
\npreWt:=x->A*x^k;" }}}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "
" 0 "" {TEXT -1 59 " Our conclusion is that weight is related to h
eight by " }{XPPEDIT 18 0 "wt=A*ht^k" "6#/%#wtG*&%\"AG\"\"\")%#htG%\"k
GF'" }{TEXT -1 90 " as determined above. We see how good our result i
s by comparing the graph with the data." }}{PARA 0 "" 0 "" {TEXT -1 0
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "J:=plot(preWt(x),x=62..
74): \nJJ:=plot(pts, style=POINT):" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 29 "with(plots):\ndisplay(\{J,JJ\});" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0
"" 0 "" {TEXT -1 89 "We use the result to predict how much a person sh
ould weigh who is 6 feet, 4 inches tall." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "preWt(76.5);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 231 "The Merc
k Manual, 14th Edition, Merck Sharp & Dohme Research Laboratories, 198
2, reports that\n\011 \011surface area = (constant)* (weight^0.42
5 )* (height^0.725).\nAlso, the surface area in square inches of a mal
e of medium built is " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "1.4
3*6^(3/2)*144;" }}}{PARA 0 "" 0 "" {TEXT -1 42 "What is the constant i
n the above formula?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "surf
ace:= h->konstant*preWt(h)^0.425*h^0.725;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 45 "solve(surface(72)=1.43*6^(3/2)*144,konstant);" }}}
{PARA 0 "" 0 "" {TEXT -1 94 "A teenage boy grew at the rate of one foo
t per year. How fast was his surface area increasing?" }}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 32 "konstant:=%;\ndiff(surface(h),h);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 13 "subs(h=70,%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0
" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }