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Homework answers / question archive / Water flows from an inverted conical tank with circular orifice at the rate dx 2?√x where r is the radius of the orifice, x is the height of the liquid level from the vertex of the cone, and A(x) is the area of the cross section of the tank x units above the orifice

Water flows from an inverted conical tank with circular orifice at the rate *dx *2?√*x*

where *r *is the radius of the orifice, *x *is the height of the liquid level from the vertex of the cone, and *A*(*x*) is the area of the cross section of the tank *x *units above the orifice. Suppose *r *= 0.1 ft, *g *= 32.1 ft/s2, and the tank has an initial water level of 8 ft and initial volume of 512(π/3) ft3. Use the Runge-Kutta method of order four to find the following:

- The water level after 10 min with
*h*= 20 s - When the tank will be empty, to within 1 min

Using Runge-Kutta method of order 4, solve problem # 28 in section 5.4 of our book. The problem statement goes like this, "Water flows from an inverted conical tank...”. You will need the formula for the volume of water in the cone, V=TR2x where x is the height (of water surface). Initially x = 8 and V=5121. You will also have to use the idea of similar triangles to compute the radius R and the cross-section A(x). Part (1) Show your steps in computing A(x). Part (2) Plug in expressions and values for pi, R, g and A(x). Write the differential equation in simplified form. Part (3) Write the initial condition, t=0, x= Solve parts (a) and (b) as given in our book. Notice that time t is in seconds. The step size h is 20 seconds. Use six decimal digits while printing final answers. Use the value of pi as 3.141593 Part(4) Book's part (a) When the book says 10 minutes, you must convert it to seconds. Print x and t at 60 second intervals. Keep six decimal digits in the answer. Part(5) Book's part (b) Print x and t at three-minute intervals to predict the time when the tank becomes empty. You have to decide about how many time-steps to take, by running your program two or three times. Part(6) Attach your computer program. If your computer program does not run, you will receive zero-credit for parts (4) and (5). 28. Water flows from an inverted conical tank with circular orifice at the rate dx dt x -0.69r²2g A(x)' where r is the radius of the orifice, x is the height of the liquid level from the vertex of the cone, and A(x) is the area of the cross section of the tank x units above the orifice. Suppose r = = 0.1 ft, g= 32.1 ft/s?, and the tank has an initial water level of 8 ft and initial volume of 512(1/3) ftº. Use the Runge-Kutta method of order four to find the following: The water level after 10 min with h = 20 s b. When the tank will be empty, to within 1 min a. Using Runge-Kutta method of order 4, solve problem # 28 in section 5.4 of our book. The problem statement goes like this, "Water flows from an inverted conical tank...”. You will need the formula for the volume of water in the cone, V=TR2x where x is the height (of water surface). Initially x = 8 and V=5121. You will also have to use the idea of similar triangles to compute the radius R and the cross-section A(x). Part (1) Show your steps in computing A(x). Part (2) Plug in expressions and values for pi, R, g and A(x). Write the differential equation in simplified form. Part (3) Write the initial condition, t=0, x= Solve parts (a) and (b) as given in our book. Notice that time t is in seconds. The step size h is 20 seconds. Use six decimal digits while printing final answers. Use the value of pi as 3.141593 Part(4) Book's part (a) When the book says 10 minutes, you must convert it to seconds. Print x and t at 60 second intervals. Keep six decimal digits in the answer. Part(5) Book's part (b) Print x and t at three-minute intervals to predict the time when the tank becomes empty. You have to decide about how many time-steps to take, by running your program two or three times. Part(6) Attach your computer program. If your computer program does not run, you will receive zero-credit for parts (4) and (5).

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