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KL
Numerade Educator

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Problem 19 Medium Difficulty

The linear density in a rod 8 m long is $ \frac{12}{\sqrt{x + 1}} $ kg/m, where $ x $ is measured in meters from one end of the rod.
Find the average density of the rod.

Answer

Average density of the rod is 6 $\mathrm{kg} / \mathrm{m}$

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Video Transcript

So the resort is over 80 m. And the density function f. of X is equal to 12 over square root of X plus one. Now we want to find the average density now because we have the density function. So we can find the mass. Uh huh. The resort will be equal to the integral From 0- eight. The density function well over a squirrel or X plus one. The eggs. Yeah. Then for this function we can do a substitution with it. You equal to square root of X plus what? Then in this case we will find D. You equal to one half. Bye one over graduate of X plus one. Then we substitute this into the pentagram. And they the upper and lower bounds when X is equal to one we will have you equal sorry when X equal to zero we will have you equal to one. And when x equal to eight we will have you equal to three. Then back to well by because do you? Sorry? Here is the X. Now because D. U. Is equal to this guy by the eggs. So we will see one over root X plus one by T. X. Should be for +22 by D. You. So here it becomes 12 by two by D. You. Then for this one is just 24 by True which is 48. So this is a total mess. Then we want to find the average mass. So the average mass. Sorry the average density average Dennis A ticket. It will be equal to the total mass divided by the total length which is 48, Divided by eight, so it's equal to six.