   Chapter 13.3, Problem 67E

Chapter
Section
Textbook Problem

# The DNA molecule has the shape of a double helix (see Figure 3 on page 850). The radius of each helix is about 10 angstroms (1 Å = 10−8 cm). Each helix rises about 34 Å during each complete turn, and there are about 2.9 × 108 complete turns. Estimate the length of each helix.

To determine

To estimate: The length of each helix in DNA.

Explanation

Given data:

Radius of helix is 10A , each helix rises about 34A , and number of complete turns are 2.9×108 .

Formula used:

Write the expression for arc length of r(t) (L) .

L=ab|r(t)|dt (1)

Here,

r(t) is first derivative of r(t) .

Consider the range of t as (0,2π) and the number of complete turns are 2.9×108 . And hence t ranges from 0 to 2.9×108×2π .

Consider a vector equation for helix as,

r(t)=rcost,rsint,zt2π

Here,

z is increment of helix, and is increases per 2π , and hence, considered as zt2π .

Substitute 10A for r and 34A for z,

r(t)=10cost,10sint,34t2π

Find the value of r(t) .

r(t)=ddt10cost,10sint,34t2π=10ddt(cost),10ddt(sint),342πddt(t)=10(sint),10cost,342π(1){ddx(sinx)=cosx,ddx(cosx)=sinx,ddx(x)=1}=10sint,10cost,342π

Find the value of |r(t)|

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