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. A spherical tank is 3 m tall and therefore has a capacity ofabout 14000 L (NOTE that 3m is its diameter, and one cubic meter is1000 L). You have been asked to put...

Question

. A spherical tank is 3 m tall and therefore has a capacity ofabout 14000 L (NOTE that 3m is its diameter, and one cubic meter is1000 L). You have been asked to put level lines on the tank showingthe depths where the tank contains 1000, 5000, and 10000 L ofliquid.(a) Use calculus to determine the height of each line (to thenearest cm)

. A spherical tank is 3 m tall and therefore has a capacity of about 14000 L (NOTE that 3m is its diameter, and one cubic meter is 1000 L). You have been asked to put level lines on the tank showing the depths where the tank contains 1000, 5000, and 10000 L of liquid. (a) Use calculus to determine the height of each line (to the nearest cm)



Answers

You are asked to design a cylindrical water tank that holds $50 \overline{0}, 000$ gal with radius $18.0 \mathrm{ft}$. Find its height $\left(1 \mathrm{ft}^{3}=7.50 \mathrm{gal}\right)$.

Here. We need to find the diameter off the tanks. So we're gonna isolate the first of all, We've been a multiplied by six over Pi to the other side to get rid off pie over six. So it's gonna be six over five multiplied by the volume, which is 15,000. That's equal to Jean too. And then, by simplifying, we're gonna have Dick, you equals to a 636.36 28,636 0.36 to find that I'm there. We do keep a crude off that that's gonna be 30.60 feet.

Yeah, so in this example they just want us to find the volume of a spherical water tank with a radius of heat meters. So we know the radius is eight m. We know the Volume for a sphere is going to be 4/3 times pi r. Cute. There's nothing else we need to do. For this example, we just plugged the eight m and for the radius in our equation, So we'll get the volume is equal to 4/3 times pi times eight meters. That's all cubed, and that will come out to 2100 44 0.6 cubic meters.

Way have a cylindrical tank and we know that the volume of a cylinder is pi r squared h and we're given that the radius in the height together have to be 50. That must mean the height is 50 minus are we also know that the volume is 40,000. So we have the equation. 40,000 equals high are square times, 50 minus are for the height. What we can do is in a graphing calculator we can set. One equation is your Why one and the other is your y two and find the intersection of them. So I'm gonna jump over and do this with dust most so one equation is 40,000. You need a wide calls by the late. And the 2nd 1 is our, um Hi. Well, minute call. Since this one dream, I'm gonna call it acts high times, pecs where I'm 50 minus x. Then we're looking for the intersection of the 2 40 thousands way up there. Way, way up. There we are. And you can zoom in the window where this is feasible. And now these air possible values for a radius and negative 14 for the radius doesn't make sense, so we're not gonna include that one. But 20.93 in 43.1 seven do so possible. Dimensions would include a radius of about 20.93 which then would have ah, height of 50 minus 29. 20.93 It should be about 29.7 or we saw the other intersection. Was that about 43.1 70 which would then have a height of about 6.83? Because that's 50 minus 43 0.17

In this problem, you're given these shown a parallel piper and were asked to determine what you dot the cross W is equal to now. In order to do that, this we have to know at least the links of you and to be and W. And so to do this. The problem tells us that the rectangular base is 300 centimeters long, in 120 centimeters wide. That means that this is equal to 300 centimeters, and this length along the side is equal to 120 centimeters. Therefore, we know that's the length of vector W. This 300 centimeters and the length of vector you is 120 centimeters. The mayor also told that the side faces are 120 centimeters, which you just sounded to you by 150 centimeters, meaning that this link is 150 centimeters. And we know over here that that is length of E. And so the length of B is equal to 150 centimeters. Now you're given a hint about how to find the cross product of two vectors or the doctor doctor to doctors. And so first we're going to try to find the cross product, and we're told to find a cross product of E. Cross W. And to do that, we're gonna take the length of the we found. That's 100 50 centimeters. I'm still length of W. That's 300 centimeters times the sine of the angle between them. And what's the angle between B and W? We know that the angle between the vertical and V is 30 degrees, and we know that the angle between B and W is the rest of that angle, which is going to be 60 degrees because the whole angle between the vertical and vector W is 90 degrees and we know 30 degrees has already taken up, so it's going to be the sign of 60 degrees. And so when we do this math, we're going to get So this is the length of the cross product via Cross W. And so when we do this math, we get 45 1000 comes a sign of 60 degrees, which is going to be Route 3/2, and if you divide 45,000 by two, we're going to get 22,000 500 Route three. Okay, And now to get the doll product between you and the vector V crossed of you, we know the following. We know that the that that dot product you don't be cross w is equal to the length of you, which we know is 120 times the length of normally. But we would have listed as one vector. But here we have enlisted as a cross product. But we're going to take the length of that cross product which we just found a Beth 22,000 500 times Route three and then multiplied by the co sign of the angle between doctor you and the cross product vector. We cross w. And that angle in this case is going to be the following. We can see that you is actually perpendicular to W and V, and we know that when you take the cross product of two vectors in this case of the cross product of W and B, you find a vector perpendicular to those vectors. And so because he was already perpendicular to them, you is going to be at the same angle as a new cross product, meaning that the angle between you and then you cross product is going to be zero degrees and co sign. It's your degrees is one. And so I'm just going to cross that out. And now we're going to take 120 times 22,000 500 and that is equal to two million 700,000. And then we had a Route three on the end, okay? And this was all part A. So, again, that is the triple product and in part B were asked for the capacity of the tank in leaders to do this. We're going Teoh, take the triple product we just got because that is the volume of the parallel pipe ed. And we know that that is in Centimeters Cube because all of our measurements were in centimeters. And so the volume waas 2,700,000 fruit, three centimeters cute. And we're told that one leader is equal to 1000 centimeters cube and said, To convert this to leaders, we're going to divide it by 1000 and we're left with if we cross out three of the zeros in the top of the bottom near left with 2700 Route three leaders, and that is our answer


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