Lab 14 Soybean Volume Assignment

Lab 14: Soybean Volume ◼ Instructions Welcome to the assignment for the Soybean Volume Lab. The purpose of this assignment is to find the volume of the soybeans in the silo described in the Lab 14 Soybean Volume introduction notebook: 1. Save the notebook to your computer right away, and save your work often. 2. When you have finished all the exercises, close any unnecessary cells and save as a PDF. (Make sure you Rasterize all 3-dimensional plots) 3. Upload the PDF on Canvas. 4. Do not delete this notebook. Save all your work until after the semester is over and your final grade has been issued. Due Date: see Canvas ◼ Volume in the second quadrant We already defined points a , b , and c as ( 0, 0, ha ) , ( 10, 0, hb ) , and ( 0, 10, hc ) , respectively in the introduc- tion notebook. We now define points d and e similarly below (go ahead and evaluate this cell): In[1]:= a = { 0, 0, ha } ; b = { 10, 0, hb } ; c = { 0, 10, hc } ; d = {- 10, 0, hd } ; e = { 0, - 10, he } ; Question 1 : Follow the procedure outlined in the Soybean Volume Lab notebook to find the equation of the plane above the second quadrant (passing through points a , c and d ). Give the equation in rectangular coordinates and write your answer in an input cell in the form “ z = . . . " In[6]:= u = d - a; v = c - a; Cross [ u, v ] Out[8]= { 10 ha - 10 hd, - 10 ha + 10 hc, - 100 } In[9]:= Solve [( 10 ha - 10 hd ) ( x - 0 ) + (- 10 ha + 10 hc ) ( y - 0 ) - 100 * ( z - ha )  0, z ] Out[9]=  z  1 10 × ( 10 ha + ha x - hd x - ha y + hc y )  Question 2 : Convert the equation of the plane in Question 1 into cylindrical coordinates and define the function p2 ( r , θ ) to be the equation of this plane. Printed by Wolfram Mathematica Student Edition

In[10]:= 1 10 × ( 10 ha + ha x - hd x - ha y + hc y ) / . x  r * Cos [ θ ] / . y  r * Sin [ θ ] Out[10]= 1 10 × ( 10 ha + ha r Cos [ θ ] - hd r Cos [ θ ] - ha r Sin [ θ ] + hc r Sin [ θ ]) In[11]:= p2 [ r _ , θ _] = 1 10 × ( 10 ha + ha r Cos [ θ ] - hd r Cos [ θ ] - ha r Sin [ θ ] + hc r Sin [ θ ]) Out[11]= 1 10 × ( 10 ha + ha r Cos [ θ ] - hd r Cos [ θ ] - ha r Sin [ θ ] + hc r Sin [ θ ]) Question 3 : To find the volume of beans in the second quadrant, we will again use a double integral in cylindrical coordinates. Write down the upper and lower limits of integration for the variable θ in the integral: Lower limit: θ = π 2 Upper limit: θ = π Question 4 : In the cell below, write the double integral that gives the volume of soybeans in the second quadrant. In[12]:=  π 2 π  0 10 p2 [ r, θ ] r  r θ Out[12]= 25 3 × ( 4 ( hc + hd ) + ha (- 8 + 3 π )) Question 5 : What you calculated in question 4 is the volume of the soybeans in the second quadrant in terms of the heights ha , hc and hd . Plug in sample values of ha = hc = hd = 30 into this volume expression. The shape corresponding to these values is a quarter of a cylinder with radius 10 and height 30. Does your answer for the volume correspond to the correct value of 750 π ? (If not, you should carefully work through questions 1-4 to see what went wrong.) In[13]:= ha = hb = hc = hd = 30; 25 3 × ( 4 ( hc + hd ) + ha (- 8 + 3 π )) Out[14]= 25 3 × ( 240 + 30 × (- 8 + 3 π )) In[15]:= π Rationalize  N  25 3 × ( 240 + 30 × (- 8 + 3 π ))  π  Out[15]= 750 π 2 Lab 14 Soybean Volume Assignment.nb Printed by Wolfram Mathematica Student Edition

In[27]:= totalVolume = 25 3 × ( 4 ( hb + hc ) + ha (- 8 + 3 π )) + 25 3 × ( 4 ( hc + hd ) + ha (- 8 + 3 π )) +  π 3 π 2  0 10 p4 [ r, θ ] r  r θ +  3 π 2 2 π  0 10 p4 [ r, θ ] r  r θ Out[27]= 1500 π + 50 3 × ( 240 + 30 × (- 8 + 3 π )) In[28]:= π Rationalize  N [ totalVolume ] π  Out[28]= 3000 π This function takes as input a list of the data measurements from the five measuring devices and gives the total volume of the soybeans in the silo. Notice that if you use 30 for each of the heights, the shape will be a cylinder with radius 10, height 30, and thus a volume of π · 10 2 · 30 = 3000 π cubic units. You should check to make sure you get the right answer from your totalVolume formula using 30 for each height. If your output was not 3000 π , then one or more of the volumes from the second, third or fourth quadrant is incorrect. Check each one, using height values of 30 for each ha , hb ,... The answer for each should be one quarter of 3000 π , or 750 π . This should help you determine which volume is incorrect. ◼ Making it useful Mathematica has the capability of pulling data from the cloud and then performing whatever calculations you like on that data. In the case of our soybean silo, let’s suppose that the five measuring devices are pushing data readings to a website where we can access them and calculate the volume of soybeans in real time. We will use the command Import t o access information from the cloud. Look up the documentation to see how it works. Question 9 : Data from the five measuring devices is stored at the following URL: http://math.mtu.edu/~jd - greger/index_files/soybean.xlsx In the input cell below, use the command Import to retrieve that data. The output should be a list containing lists with 5 elements. These correspond to the data readings from the measuring devices In[29]:= Import [ "http: // math.mtu.edu /~ jdgreger / index _ files / soybean.xlsx" ] Out[29]= {{{ 19.5646, 19.0045, 19.1209, 19.6134, 18.5936 } , { 18.0135, 17.3152, 17.7231, 17.3632, 19.9631 } , { 17.2857, 19.8853, 17.6645, 19.4786, 16.4958 } , { 16.919, 16.252, 17.3649, 16.5916, 19.487 } , { 16.175, 19.2394, 19.6143, 16.2583, 16.9318 } , { 17.0605, 19.5122, 18.4925, 16.1222, 17.773 } , { 16.1675, 19.2995, 18.3258, 16.6761, 16.6718 }}} 4 Lab 14 Soybean Volume Assignment.nb Printed by Wolfram Mathematica Student Edition

Question 10 : Use your volume formula from the previous section to find the volume of the soybeans in the bin at the very first 5-element list from the data you just imported. In[30]:= ha = 19.564593387855407; hb = 19.00445918370692; hc = 19.120939083950873; hd = 19.613418642567765; he = 18.593638616085077; In[35]:= totalVolume2 = 25 3 × ( 4 ( hb + hc ) + ha (- 8 + 3 π )) + 25 3 × ( 4 ( hc + hd ) + ha (- 8 + 3 π )) +  π 3 π 2  0 10 p3 [ r, θ ] r  r θ +  3 π 2 2 π  0 10 p4 [ r, θ ] r  r θ Out[35]= 7738.97 Lab 14 Soybean Volume Assignment.nb 5 Printed by Wolfram Mathematica Student Edition