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Problem 53 Easy Difficulty

The DNA molecule has the shape of a double helix (see
Figure 3 on page 582 ). The radius of each helix is about
10 angstroms $\left(1 \mathrm{A}=10^{-8} \mathrm{cm}\right)$ . Each helix rises about
34 A during each complete turn, and there are about
$2.9 \times 10^{8}$ complete turns. Estimate the length of each
helix.

Answer

2 $\mathrm{m}$

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Video Transcript

{'transcript': "So we have the equation of one of the helix is, um it's gonna have the form a co sign p a sigh Inti bt where the constants and B will be determined from the given data since the radius of the helix is about and and a So this is you know, that this right here is equal to 10 to the negative eighth centimeters. So now what we end up having with 10 a. This is going to actually give us a 10 to the negative seventh centimeters. So based on that, we have, um, in one period, Z coordinate rises by two pi b. So we get that two pi b equals 34. Okay, so we saw for B. We end up getting 5.41 times 10 to the negative eighth centimeters. Then we conclude that the equation of one of the hell asses is going to be 10 Kasai, Inti negative 10 Sai Inti and 17 over pi t. And then the equation of the other helix is gonna be negative. 10 ko 70 because there double helix. So they're gonna be almost opposite of each other. Um, this was not negative before but this will be negative. Negative. 10. Sigh Inti and this will be still positive. 17 over Pie Team, then the length of the Helix heels. These are the same, so we can just compute one of them. And that's going to be the integral from 0 to 2 n pie of the square root of a co sign t prime squared plus a sign t prime squared Class B E T Prime Square And what? And that's D T. So what we're gonna end up getting as a result, is to and pie times the square root of a squared plus B squared. Since there are about 2.9 times 10 to the eighth complete turns, what we end up getting is we can, um, add our values in and plug them in. What will end up having is that l is equal to two times 2.9 times 10 to the eighth because that's our number of cycles, and then we want pie. Then we multiply this times the square root of our 10 a squared plus our 17 over pi a square, and this is going to give us approximately 2 m"}

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