Question 1163697
If {{{z=k*d^n}}} , then {{{log(10,z)=log(10,k)+n*log(10,d)}}}
The graph of {{{log(10,z)}}} against {{{log(10,d)}}} would be expected to be a straight line with slope {{{n}}} up to a certain value {{{log(10,d)=log(10,D)}}} .
We need to calculate and tabulate {{{log(10,z)}}} and {{{log(10,d)}}}
{{{matrix(9,4,d,z,log(10,d),log(10,z),750,2.1,2.875061263,0.322219295
,810,2.6,2.908485019,0.414973348,870,3.2,2.939519253,0.505149978,930,4.0,2.968482949,0.602059991,990,4.8,2.995635195,0.681241237,1050,5.6,3.021189299,0.748188027,1110,5.9,3.045322979,0.770852012,1170,6.1,3.068185862,0.785329835)}}}    
 
a)Then we plot {{{log(10,z)}}} against {{{log(10,d)}}}
{{{drawing( 300, 300, 2.8, 3.1,0.25,0.85,
graph( 300, 300, 2.8, 3.1,0.25,0.85,0.6332x-1.1576,2.9634x-8.2005),
arrow(2.81,0.55,2.81,0.2),arrow(2.81,0.55,2.81,0.85,0.45),
line(2.81,0.35,2.805,0.35),line(2.81,0.45,2.805,0.45),
line(2.81,0.55,2.805,0.55),line(2.81,0.65,2.805,0.65),
line(2.81,0.75,2.805,0.75),locate(2.815,0.77,0.75),
locate(2.815,0.67,0.65),locate(2.815,0.57,0.55),
locate(2.815,0.47,0.45),locate(2.815,0.37,0.35),
arrow(2.95,0.27,2.7,0.27),arrow(2.95,0.27,3.1,0.27),
line(2.85,0.275,2.85,0.265),line(2.9,0.275,2.9,0.265),
line(2.95,0.275,2.95,0.265),line(3,0.275,3,0.265),
line(3.05,0.275,3.05,0.265),locate(2.835,0.305,2.85),
locate(2.885,0.305,2.90),locate(2.935,0.305,2.95),
locate(2.985,0.305,"3.00"),locate(3.035,0.305,3.05),
green(circle(2.875,0.3222,0.005)),green(circle(2.908,0.415,0.005)),
green(circle(2.94,0.5051,0.005)),green(circle(2.968,0.6021,0.005)),
green(circle(2.996,0.6812,0.005)),green(circle(3.021,0.7482,0.005)),
red(circle(3.045,0.7709,0.005)),red(circle(3.068,0.7853,0.005))
)}}} The points in green, up to {{{log(10,d)=3.021}}} , corresponding to {{{d=1050}}} fit well enough on a line. That supports suggesting {{{highlight(D=1050)}}} .
 
b) We could estimate {{{n}}} as the slope between points {{{"("}}}{{{log(10,d)}}}{{{","}}}{{{log(10,z)}}}{{{")"}}} {{{"("}}}{{{2.875}}}{{{","}}}{{{0.3222}}}{{{")"}}} and {{{"("}}}{{{3.021}}}{{{","}}}{{{0.7482}}}{{{")"}}}
That slope is {{{(0.7482-0.3222)/(3.021-2.875)=0.4260/0.146=2.92("rounded")}}}
Since {{{n}}} is supposed to be a whole number, it must be {{{highlight(n=3)}}}.


c) For {{{k=5×10^-9}}} {{{log(10,k)=log(10,5*10^-9)=-8.30103}}} (rounded)

Substituting values for {{{z=3.0}}} , {{{n=3}}} {{{k=5×10^-9}}}, with {{{log(10,k)=-8.30103
}}} and {{{log(10,3)=0.47712}}} into {{{log(10,z)=log(10,k)+n*log(10,d)}}} we get
{{{0.47712=-8.30103+3log(10,d)}}}
{{{0.47712+8.30103=3log(10,d)}}}
{{{8.77815=3log(10,d)}}}
{{{8.77815/3=log(10,d)}}}
{{{2.92615=log(10,d)}}}
{{{d=10^2.92615}}} to get {{{highlight(d=843)}}} as the value of d for which z=3.0.