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Math
CalculusMultivariable Calculus8th Edition

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Chapter 13.3, Problem 67E
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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Problem 50E
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Problem 65E
Problem 66E
Problem 67E
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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
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 10 A∘ , each helix rises about 34 A∘ , 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,

r is radius,

z is increment of helix, and is increases per 2Ï€ , and hence, considered as zt2Ï€ .

Substitute 10 A∘ for r and 34 A∘ for z,

r(t)=〈10cost,10sint,34t2π〉

Find the value of r′(t) .

r′(t)=ddt〈10cost,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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Sect-13.1 P-1ESect-13.1 P-2ESect-13.1 P-3ESect-13.1 P-4ESect-13.1 P-5ESect-13.1 P-6ESect-13.1 P-7ESect-13.1 P-8ESect-13.1 P-9ESect-13.1 P-10E
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