Problem:
The DNA molecule has the shape of a double helix. The radius of each
helix is about 10 angstroms (1 angstrom = 10^(-8) cm). Each helix rises
about 34 angstroms during each complete turn and there are about
2.9 x 10^(8) complete turns. Estimate the length of each helix.
|
| and as usual the i,j,k basis vectors: |
> alias(v=vector): i:=v([1,0,0]): j:=v([0,1,0]): k:=v([0,0,1]):
| The radius of the helix is a, and the hight of each turn is h, in angstroms. |
> a := 10: h := 17/Pi:
| The vector function for this helix is then, r |
> r := a*cos(t)*i + a*sin(t)*j + h*t*k;
t k
r := 10 cos(t) i + 10 sin(t) j + 17 ---
Pi
| The velocity vector is: |
> V := diff(r,t);
k
V := -10 sin(t) i + 10 cos(t) j + 17 ----
Pi
| from where we obtain the speed at time t by computing the length of V. The length of one complete revolution is then the integral of the speed, |
> L := Int(len(evalm(V)),t=0..2*Pi);
2 Pi
/ 2 1/2
| (100 Pi + 289)
L := | ------------------ dt
| Pi
/
0
| which evaluates to: |
> L := evalf(L);
L := 71.44117693
| This is given in angstroms. The total length of 290 million revolutions is (in centimeters): |
> TotL := 2.9*L;
TotL := 207.1794131
| i.e. about 2 meters long! |