<Text-field style="_cstyle259" layout="Heading 1"><Font bold="true">Volumes That You Can Use</Font></Text-field> Note: You may notice differences between this Maple worksheet and the equivalent Mathematica notebook. These differences were introduced to preserve the content of these modules and were necessary because of major functional differences between Maple and Mathematica.
<Text-field style="_cstyle256" layout="Heading 1"><Font size="18">Introduction</Font></Text-field> OBJECTIVE: Use the concept of volume to solve practical problems involving rain catchers and satellite dishes. How can you measure precipitation if you collect rain in a container that is narrower at the bottom than at the top? At what angle must a satellite dish be tilted for rain not to collect in it?
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<Text-field style="Heading 1" layout="Heading 1">Part I: A Hemispherical Rain Gauge </Text-field> Measuring Rainfall with a Hemisphere Suppose you are trying to determine how to use the height of water in a hemispherical bowl of radius and depth 10 inches to measure the amount of rain that has fallen. Why do we use the volume of a cylinder of radius 10 inches filled to a particular height (k)? What does that have to do with the volume of water that will collect in the bowl? The key idea is that the volume in each is the same; the only difference is in the height of the water in the two different containers. We will write our formulas using a general radius (ra), so that we can go back and change it out for hemispherical bowls with different radii. Since the bowl is considered to be sitting on top of the xy-plane with the axis of symmetry through the z axis, it is easiest to use cylindrical coordinates. The equation of the sphere is 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. restart; volcylinderht:= Pi*ra^2*k: volbowlht:=int(int(int(r,z=(ra-sqrt(ra^2-r^2))..h),r=0..sqrt(2*h*ra-h^2)),theta=0..2*Pi): print(`volume in cylinder filled to height k is `, volcylinderht); print(`volume in bowl filled to height h is `, volbowlht); NiRJSnZvbHVtZX5pbn5jeWxpbmRlcn5maWxsZWR+dG9+aGVpZ2h0fmt+aXN+RzYiKihJI1BpRyUqcHJvdGVjdGVkRyIiIkkjcmFHRiQiIiNJImtHRiRGKA== NiRJRnZvbHVtZX5pbn5ib3dsfmZpbGxlZH50b35oZWlnaHR+aH5pc35HNiIsMiooLUklY3Nnbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSNyYUdGJCIiIkYtIiIkSSNQaUdGKkYuIyIiI0YvKiotRig2IywmSSJoR0YkRi5GLSEiIkYuRi1GMkY3Ri5GMEYuISIjKipGNEYuRjdGMkYtRi5GMEYuRjIqKEY0Ri5GN0YvRjBGLiNGOUYvKihGNEYuRi1GL0YwRi5GMSooRjdGMkYtRi5GMEYuRi8qKEYtRjJGN0YuRjBGLkY5KiZGN0YvRjBGLkY4 If there is 1 inch of rain in the 10-inch radius cylinder, we will determine the height of water in the bowl. Why do we equate the volume in the cylinder to the volume in the bowl? volc:=subs({ra=10, k=1},volcylinderht): volb:=subs(ra=10,volbowlht): one:=fsolve(volb=volc,h); JCIrdm1cYkwhIio= print(`If there is one inch of rain in the 10 - inch radius cylinder, the hemisphere will be filled to the `, one, `inch level`); NiVJX3FJZn50aGVyZX5pc35vbmV+aW5jaH5vZn5yYWlufmlufnRoZX4xMH4tfmluY2h+cmFkaXVzfmN5bGluZGVyLH50aGV+aGVtaXNwaGVyZX53aWxsfmJlfmZpbGxlZH50b350aGV+RzYiJCIrdm1cYkwhIipJK2luY2h+bGV2ZWxHRiQ= What if there are 3 inches of water in the bowl? How many inches of water would this produce in the cylinder? volc:=subs(ra=10,volcylinderht): volb:=subs({ra=10, h=3},volbowlht): three:=fsolve(volb=volc): print(`If there are three inches of rain in the bowl, the actual precipitation level is `, three, `inches`); NiVJXHBJZn50aGVyZX5hcmV+dGhyZWV+aW5jaGVzfm9mfnJhaW5+aW5+dGhlfmJvd2wsfnRoZX5hY3R1YWx+cHJlY2lwaXRhdGlvbn5sZXZlbH5pc35HNiIkIissKysrIikhIzVJJ2luY2hlc0dGJA== Because of multiple roots, it is somewhat difficult to solve explicitly for the height in the bowl as a function of the actual amount of rain (height of water in the cylinder). However, it is easy to determine the relationship the other way around; that is, if we know the height of water in the bowl, we can easily find the amount of precipitation by dividing the volume of water in the bowl by 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, which is the cross-sectional area of the cylinder with the same size top. That computes the actual precipitation amount. We do this for a general radius and look at some tables of values associating the actual precipitation and the height of the water in the bowl corresponding to different values of the radius. The formulas below simply show that the height in the cylinder, representing actual precipitation, is simply the volume of collected water divided by the cross-sectional area of the cylinder 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. precip:=volbowlht/(ra^2*Pi); KigsMiooLUklY3Nnbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSNyYUdGKiIiIkYsIiIkSSNQaUdGKEYtIyIiI0YuKiotRiY2IywmSSJoR0YqRi1GLCEiIkYtRixGMUY2Ri1GL0YtISIjKipGM0YtRjZGMUYsRi1GL0YtRjEqKEYzRi1GNkYuRi9GLSNGOEYuKihGM0YtRixGLkYvRi1GMCooRjZGMUYsRi1GL0YtRi4qKEYsRjFGNkYtRi9GLUY4KiZGNkYuRi9GLUY3Ri1GLEY4Ri9GNw== We can apply these formulas to create a list of values that associate the height of water in the bowl with the actual precipitation. Digits:=20: precip1:=subs(ra=10, precip): seq([0.5*i, subs(h=0.5*i, precip1)], i=1..20): preciplist1:=evalf(%): [`10" bowl level `, ` actual precip.`]; matrix([preciplist1]); p1:=plots[pointplot]({preciplist1}, labels=["height","precipitation"], color=blue): #print(p1); NyRJMzEwIn5ib3dsfmxldmVsfn5+fkc2Ikkzfn5+fmFjdHVhbH5wcmVjaXAuR0Yk 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 precip2:=subs(ra=15, precip): seq([0.5*i, subs(h=0.5*i, precip2)], i=1..20): preciplist2:=evalf(%): [`15" bowl level `, ` actual precip.`]; matrix([preciplist2]); p2:=plots[pointplot]({preciplist2}, labels=["height","precipitation"], color=green): #print(p2); NyRJMjE1In5ib3dsfmxldmVsfn5+RzYiSTJ+fn5hY3R1YWx+cHJlY2lwLkdGJA== 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 precip3:=subs(ra=20, precip): seq([0.5*i, subs(h=0.5*i, precip3)], i=1..20): preciplist3:=evalf(%): p3:=plots[pointplot]({preciplist3}, labels=["height","precipitation"], color=red): #print(p3); [`20" bowl level `, ` actual precip.`]; matrix([preciplist3]); NyRJMjIwIn5ib3dsfmxldmVsfn5+RzYiSTJ+fn5hY3R1YWx+cHJlY2lwLkdGJA== 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 print(plots[display]({p1,p2,p3})); 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 Which plot goes with which radius?
<Text-field style="Heading 1" layout="Heading 1">You Try It: Part I</Text-field> Now, select a different shaped rain-collector, and determine the relationship between the height of water in your collector and the actual precipitation. Suppose that the bowl is parabolic, say with the equation 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 ,with maximum radius 10 inches. Digits:=8: volofbowlht:=int(int(int(r,z=.5*r^2..h),r=0..sqrt(2*h)),theta=0..2*Pi); LCQqJEkiaEc2IiIiIyQiKUZmVEohIig= You can enter ra = 10 at the start and replace the volofbowlht entry with the above and proceed. The relationship is much simpler here. Give the precipitation level for height h in your bowl as follows. ra:=10: newprecip:=simplify(volofbowlht/(ra^2)*Pi); LCQqJEkiaEc2IiIiIyQiKVpncCkqISIq Find the precipiation level if the height in the bowl is 2.5 inches. subs(h=2.5, newprecip); JCIpSF1vaCEiKQ== Repeat these steps for containers having different shapes.
<Text-field style="Heading 1" layout="Heading 1">Part II: Water in a Satellite Dish</Text-field> The Tilt of a Satellite Dish As we tilt the bowl, it is convenient to use rectangular coordinates and picture the tilted bowl upright and show the water as tilted. We will let the bowl be centered on the z axis and rest on top of the xy-plane. The equation of the plane that represents the water level can be written as z = m y + b. In each case, since the water level plane must pass through the coordinate points z = 0.5 when y = 1, the intercept b can be written as 0.5 - m. xyzbowl:=(x,y)->.5*(x^2+y^2); xyzplane:=(y,m)->m*y+(.5-m); plotbowl:=plot({xyzbowl(0,y), xyzplane(y,0)}, y=-1..1, color=[blue, cyan]): plot1:=plot(xyzplane(y,.25), y=-.5..1, color=green): plot2:=plot(xyzplane(y,.5),y=0..1, color=red): print(plots[display]({plotbowl, plot1, plot2})); Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYkIiImISIiIiIiLCYqJDkkIiIjRi4qJDklRjJGLkYuRiVGJUYl Zio2JEkieUc2IkkibUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJjklIiIiOSRGLUYtJCIiJiEiIkYtRixGMUYlRiVGJQ== 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Let us begin by drawing some scenarios as the slope of the plane (value of m) goes from 0 to .75. We can integrate to find the volume in each case first by finding the boundary curve for x and y along which the plane and the paraboloid intersect and then by leaving our integral written as a function of the slope of the water level plane. sol:=solve(xyzplane(y,m)=xyzbowl(x,y),y); sol1:=unapply(sol[1], (x,m)); sol2:=unapply(sol[2], (x,m)); NiQsJkkibUc2IiIiIiokLCoqJEYkIiIjRiYkRiYiIiFGJkYkJCEiI0YsKiRJInhHRiVGKiQhIiJGLCNGJkYqRiYsJkYkRiZGJ0Yx Zio2JEkieEc2IkkibUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCY5JSIiIiokLCoqJEYrIiIjRiwkRiwiIiFGLEYrJCEiI0YyKiQ5JEYwJCEiIkYyI0YsRjBGLEYlRiVGJQ== Zio2JEkieEc2IkkibUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCY5JSIiIiokLCoqJEYrIiIjRiwkRiwiIiFGLEYrJCEiI0YyKiQ5JEYwJCEiIkYyI0YsRjBGN0YlRiVGJQ== Note that the equation of intersection of the plane and the paraboloid is a circle, centered at x = 0 and y = m and with radius 1 - m, so we will let z go from the paraboloid to the plane and integrate x and y over the appropriate circle. The following command will define the volume as an integral. The integral itself will be evaluated for selected values of m. int(int(int(1, z=xyzbowl(x,y)..xyzplane(y,m)), y=sol[1]..sol[2]), x=(m-1)..(1-m)): vol:=unapply(%,m); Zio2I0kibUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQsLiomOSQiIiIsJiokLCZGMUYyKiQsKiokRjEiIiNGMiRGMiIiIUYyRjEkISIjRjsqJEkieEdGJUY5JCEiIkY7I0YyRjlGQEY5RjIqJCwmRjFGMkY2RjJGOUZARjIkIikrKytdISIpRjZGQComRjFGMkY3RkIkRjlGOyomRj9GOUY3RkJGMiokRjUiIiQkISlubW07RkcqJEZERkwkIilubW07RkcvRj87LCZGMUYyRkFGMiwmRjJGMkYxRkFGJUYlRiU= Now, let's visualize the results for a few values of m. We first read in a package to help us plot. Then we let the slope take on the values of .25, 1, and .75, each time evaluating the volume of water in the satellite. plot({xyzplane(y,0),xyzbowl(0,y)}, y=-1..1, labels=[y,z], title="Slope of Water is 0"); print(`When the slope is 0, the volume of the water is `, abs(vol(0)),`cubic meters.`); plot({xyzplane(y,.25),xyzbowl(0,y)}, y=-.5..1, labels=[y,z], title=`Slope of Water is .25`); print(`When the slope is .25, the volume of the water is `, abs(vol(0.25)),`cubic meters.`); plot({xyzplane(y,.5),xyzbowl(0,y)}, y=0..1, labels=[y,z], title="Slope of Water is .5"); print(`When the slope is .5, the volume of the water is `, abs(vol(.5)),`cubic meters.`); p3:=plot({xyzplane(y,.75),xyzbowl(0,y)}, y=.5..1, labels=[y,z], title="Slope of Water is .75"): p4:=plot({xyzbowl(0,y)}, y=0...1, labels=[y,z], color=green): print(plots[display]({p3,p4})); print(`When the slope is .75, the volume of the water is `, abs(vol(.75)),`cubic meters.`); 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 NiVJUVdoZW5+dGhlfnNsb3BlfmlzfjAsfnRoZX52b2x1bWV+b2Z+dGhlfndhdGVyfmlzfkc2IiQiKTspUiZ5ISIpSS5jdWJpY35tZXRlcnMuR0Yk 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<Text-field style="Heading 1" layout="Heading 1">You Try It: Part II</Text-field> Check out satellite dishes that are installed at residences or at communication facilities. See if their shape is parabolic, and find out what their tilt is. Does it match up to what you expected?