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227. Piles are commonly used in foundations of civil work structures. Test piles were driven and used to measure pile strength at a selected site: The following str...

Question

227. Piles are commonly used in foundations of civil work structures. Test piles were driven and used to measure pile strength at a selected site: The following strength data (in kips) were collected: 8829,10236,5101,9144,7790, 9327, 8470,10570,10186, 9305,10746,9069,11436,10044,10281,12311, 10639, 10215,8723,and 8877. Plot the histogram and frequency diagrams for pile strength.

227. Piles are commonly used in foundations of civil work structures. Test piles were driven and used to measure pile strength at a selected site: The following strength data (in kips) were collected: 8829,10236,5101,9144,7790, 9327, 8470,10570,10186, 9305,10746,9069,11436,10044,10281,12311, 10639, 10215,8723,and 8877. Plot the histogram and frequency diagrams for pile strength.



Answers

In Figures $28.4$ and $28.5$, let $R_{1}=11.0 \Omega$, let $R_{2}=$ $22.0 \Omega$, and let the battery have a terminal voltage of $38.0 \mathrm{~V} .$ (a) In the parallel circuit shown in Figure $28.5$, which resistor uses more power? (b) Verify that the sum of the power $\left(I^{2} R\right)$ used by each resistor equals the power supplied by the battery $(I \Delta V) .(\mathrm{c}) \mathrm{In}$ the series circuit, which resistor uses more power? (d) Verify that the sum of the power $\left(I^{2} R\right)$ used by each resistor equals the power supplied by the battery $(\mathscr{P}=I \Delta V)$. (e) Which circuit configuration uses more power?

Welcome to you Marie in the current problem we are considering the distribution of voltages of double A batteries and this voltages are distributed normally. That mean 1.50 votes and sigma 0.5 holes. Now, if we draw the normal distribution over here, that will look something like this. Okay, but this is the mean. Okay, this is the point of symmetry. Uh consider my beautiful drawing here and this is one sigma limit. This is two sigma limit and this is three sigma limit. Now the question that we are asked is what percentage of the uh written Batteries will have voltages above? 1.64, correct. So now 1.64 is again a non standard value. It's not one sigma tau sigma three sigma limit value. So we take the process of standardization x minus mu pi sigma greater than 1.64 minus New. New is 1.50 divided by 0.05, correct. Which is equal to the probability said greater than 0.14 divided by zero 05 Date it is probability zed greater than 14 x five. So this will be said greater than 2.8. Now, what is the value of 2.8 from the tables? So if I bring the table in this will be always um so this will be always 2.8 over here, which is 0.9974. So below, if if we look at the diagram now, So we want to .8 which is 1.64. So 1.64 is going to be somewhere forward. Fear. If you consider it's under stable, the probability of Now this stable gives probability zed less than 2.8 which is 0.9974 But we want greater than so, it will be one minus probability zero less than 2.8. That is one minus zero point 9974 Which is equal to six and then two. And then you know that you know They do 0.0026. But uh probability if we can get that 2%, it it will be 0.26% by multiplying this 100 right? There's little .26% of the entire population. As as you can see like if you ask yourself why why is this list? Because to pointed if you see 1.64 is the value at 1.65 you get the three signal a bit. So just really a very rare value. So something more than that rare value. We have really negligible probability. That is a very negligible percent of the population will go and sit there. So only this percentage will go and sit in this area and

Okay, So for this problem we have this took a few good nations resistance. We have the same values of the resistance, but one is in serious. One is in parallel. Onda, we have the same potential difference or voltage. So I used this color coal on in the first part. We want to know which one off the resistance delivers more power. Okay, so this is easy. Remember that Voltages territory. Both. Okay. And we know that power equals both that where divided by the resistance. So obviously, since this is in parallel, the vaulter it is the same on the power will depend on the value of the rest. And for a larger value, there will be less power. But let's calculate. So for the first resistance, we have a 33 were divided by 11. Okay, And this is going to be 99 1. And for the second one, this is going to be very, really squared, but divided in between 22. So this is going to be 49.5 watts and then the first one and has worked. Yeah, so then we're asked to find that the some of these two powers equals the total power. Thanks. So for this, we know that p one plus p two is going to be 99 plus 49.5. This is 100 48.5. What? So we want to know this is the total power. Okay, so too calculated. Using the total values of the circuit, we will have to find the current. So we know that current big balls voltage divided by the resistance. So this is going to be very tree divided by Remember that this is a, uh, Caroline. So one over one divided by 11 plus one divided by 22. If you make these calculations is not really hard. You're going to find that that the current is 4.5. Okay. And then you sent this. We know that power or total power is going to be, um I'm sorry. 4.5 divided, but multiplied by territory. Votes on this is remember, this is also current. What supplied by the currency? This is 148.5 watts. You can check the calculations yourself, so yes, we already better. And we are asked SRE to do the same. But now for the seriously with Okay, So which one delivers more power again? It's, uh, dependent. We can not use the same formula because we know that for a serious you're great. Uh, the but then shall, uh, yeah, or the world. And it depends on the results. Okay, so we're going instead to use that power equals the current squared times the resistance. Okay, so we know that the total current is going to be the voltage divided by the total resistance. So this is going to be very three divided by the total resistance is going to be 11 lost, 22. Okay, so this is actually 33 divided by territory, and this is one. So this is the total current, and it's going to be the same in each of the resistance because it's a serious. Okay, So for the first one, then I have one square want supplied by 11. So I have 11. And for the second gun, one is one squared. Want to play by 22. So it's going to be 20 to 1. So, for this one, actually, the other resistance is the one who has more power. Okay, so for Syria's are, too. The liver's more power and parallel is the country nights Here are one delivers more And now I am asked the same as before to verify that the some of the power it's the total power. Okay, so if I sum this and going to have total power equals 11 once plus 22 once this is very +31 mhm and our yes, uh, you know, intensity of the current want to fly by the potential difference. So this is one unfair. I remember it all power. Want to play by directory that it's the difference important. And this is the retreat one, which is actually the same. Okay, so it's already better fied on. The last question is which configuration? Siri's or parallel delivers more power. Okay, so I know the total power for Siri's is there is three. What? And it's all power for parallel. You can check it. Uh, here it's 148.5, 148. So clearly the parallel configuration is this more or deliver smallpox

In the first part of this problem we have to find the batteries internal resistance that is generated by a smaller. Let's write the equation for the first case. So the case one we can wipe the M. F. As is equal to I won smaller plus I've won capital of but this uh we want is equal to I wan. Capital of this is the potential drop across the resistor. Uh that set the values into these questions. So here we have e he's equals two. The island which is equal to two point five ft of bed into odd plus. Into 2.5 ft in bed into capital are which is equals to four point five to my God. So from here we can wipe this question as is equals two 2.54 and bad into smaller plus. Here we will get 11 point 481 uh fault. So this will be a request number one. Mhm. Now for guests to we can white the IMF of the battery as equals two into 4.98 And bad into smaller plus uh 4.98 in bed. Into the resistance to point to one omega. This is the external instance for the guest too. So from here we write we write the question as 4.98 and bad into smaller plus 11.6 Work. And this should be our equation number two. Yeah. Yeah. Now using equation number one and number two we can write the value for this are smaller as a smaller is equals two 0.19 Uh So this is the value for the internal resistance of the battery. In the part of this problem we have to calculate the IMF of the battery that is E. In order to calculate this E M F. We can use that equation about one. Uh equally number two of the Kiss. What? So this is the values into going number one. So easy equals two 2.45 And bad into the value for the smaller, which is equals to 0.19 omega plus 11 point for it. One looked. So we will get the mm for the battery as a 11.95 fold. Move to Patsy. So in the party we have to calculate the terminal voltage for both cases. So we need to calculate we won and the turn feature so we can use their own flyers. We one is equal to Ivan are one and this future will be equals two. I too. Uh so that everyone will be equals two. I want which is equals two 2.5 ft and bed into our one which is equals to 4.5 true mega. So we went with B equals two 11 point for it. Vote. Generally we can wait for me to as we, too is equals two I, too. That is given by 4.98 compared than to 2.21 omega. So we too, will be equals to 11.1 world. Yeah. Thank you.

Welcome to enumerate in the current problem. We have the voltages of aa batteries, mhm of a batteries. And the variable is distributed with mean this 1.50 and Sigma 0.05 goals. Now we have to find the probability of what percentage of the batteries will lie between 1.52 To 1.58. Now we we cannot find this using uh empirical table because for that we will need this exactly what six that we can find. But if it is particular points like this, which are not on particular sigma levels, then we To a standardization. So 1.5 to -1.5 divided by 0.05, Less than it was two. X -1.5 divided by 0.05. Less than equal to I will write down Who? -1.5 divided by 0.05. So if we simplify that, we will get 0.02 Divided by 0.05. Less than said, less than over here. It will be zero 08 divided by 0.0 Fight, correct? This can be written as probability two by five. Less than said. Less than eight by fight. Which is equals two, probability 0.4. Less than zero, less than one point six. Well now, if we bring the standard normal table over here, treat me this, but I will just it used the size. So this will be Probability of zed less than 1.6 minus probability. Zeb less than zero food. What do we mean by that? See if this is a distribution and we want the values off. See Z equals to zero four, correct. 0.4. So we will read that Valium I'm here. So you see it's 0.6554. So it will be this area will be 0.6554. So let us write 0.6554 and 1.6 will be somewhere over here. So that will have a bigger area, correct? So if you go and see 1.6 and 0.945 to 0.9452 therefore, if we subtract, we get eight and then nine and then eight and then 892 So this is the probability. So the percentage will be 28.98%, which can be rounded up to 23%. So 23% of the Batteries will be having a lifetime, uh will be having a four days between 1.52 and 1.5. It so I hope you could understand this. Let me know if you have any questions.


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