Volumes of Solids of Revolution and Arc Length Name:

Example 1: The region bounded by the curve 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 and the x -axis between x=0 and x=4 is revolved about the x -axis. Find the volume of the solid which is formed. Solution: As usual it is a good idea to make some plots in order to visualize the problem. First define the function 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 . restart;with(plots): f :=x->x*exp(-x); The next Maple segment plots the graphs of the function, the line segment from (4,0) to (4,f(4)), and a typical element of area for the region under consideration. p1:=plot(f(x),x=0..4): p2:=plot([[4,0],[4,f(4)]]): e1:=polygonplot([[1,0],[1.2,0],[1.2,f(1.2)],[1,f(1)]],color=green): t1:=textplot([1.1,.39,convert([68],bytes)],font=[SYMBOL,12]): t2:=textplot([1.2,.39,`x`]): display({p1,p2,e1,t1,t2}); The solid is formed by revolving the region about the x-axis will now be plotted. An element of volume is obtained by revolving an element of area for the region about the x-axis. The solid and a typical element of volume will be made. The Maple procedure called tubeplot is used. The procedure is part of the plots package. The proper syntax for it is tubeplot(C, options ), where C is a set of space curves. The radius is assigned to be f(x) for the tubeplot of the entire solid. radius:='radius':with(plots): p4:=tubeplot([x,0,0],x=0..4,radius=f(x),axes=normal): display(p4,orientation=[-60,75]); One can think of approximating the volume of the solid by adding up the small cylinders with radius equal to f(x) and thickness 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. Such a cylinder has volume 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. Terms like this appear as a summand in the sum obtained by partitioning the interval [0,2] and writing a Riemann Sum. Therefore, the volume is V = 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. Calculate the volume exactly, as follows: v:=Pi*Int(f(x)^2,x=0..4); v1:=value(v); The volume to ten digits of accuracy is equal to evalf(v1); The next example illustrates that sometimes the element of volume can be hollow. Example 2: Find the volume of the solid obtained by rotating the region in the first quadrant bounded by the curves 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 and y = x about the x -axes. restart;with(plots): Solution: First find the x value of the point of intersection. f:=x->2*sin(x):g:=x->.5*x; s:=fsolve(f(x)=g(x),x=0..Pi); The following Maple segment plots the region bounded between the curves along with a typical element of area. rect:=polygonplot([[0.9,g(1)],[1.1,g(1)],[1.1,f(1)],[0.9,f(1)]],color=blue): p1:=plot({f(x),g(x)},x=0..2,thickness=2): t1:=textplot([.95,1.75,convert([68],bytes)],font=[SYMBOL,14]): t2:=textplot([1.0,1.75,`x`],font=[HELVETICA,12]): display({p1,rect,t1,t2}); Now rotate the element of area about the x -axis to create a cylindrical solid with a hole drilled through the center. The following plot produces a cross-sectional view of this cylinder as viewed looking down the positive x -axis. implicitplot({y^2+z^2=g(1)^2,y^2+z^2=f(1)^2},y=-2..2,z=-2..2, scaling=constrained,thickness=2); Note that the area of a typical cross-section is equal to the area of the larger circle minus the area of the smaller circle: 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. This means that an element of volume is NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYoLyUqbWF0aGNvbG9yR0ZELyUvbWF0aGJhY2tncm91bmRHRkQvJStmb250ZmFtaWx5R0YyLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGKC8lKW1hdGhzaXplR0Y1LUYkNidGLC1GJDYlLUYtNjlRJkRlbHRhRihGMEYzRjYvRjpGOEY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbi9GaW5RJ25vcm1hbEYoRltvLUkjbW9HRiU2M1ExJkludmlzaWJsZVRpbWVzO0YoLyUlZm9ybUdRJmluZml4RigvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGOC8lJ2xzcGFjZUdRJDBlbUYoLyUncnNwYWNlR0ZkcC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobWF4c2l6ZUdRKWluZmluaXR5RigvJShtaW5zaXplR1EiMUYoLyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJTBmb250X3N0eWxlX25hbWVHRlcvJSVzaXplR0Y1LyUrZm9yZWdyb3VuZEdGRC8lK2JhY2tncm91bmRHRkQtRi02OVEiVkYoRjBGM0Y2RjlGPEY+RkBGQkZFRkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm5GaG5GW28tRmhvNjNRIj1GKEZbcEZecEZgcC9GY3BRL3RoaWNrbWF0aHNwYWNlRigvRmZwRmZyRmdwRmlwRltxRl5xRmFxRmNxRmVxRmdxRmlxRltyRl1yLUYkNiktRi02OVEjUGlGKEYwRjNGNkZkb0Y8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZlb0Zbb0Znby1GJDYlLUZobzYzUSIoRigvRlxwUSdwcmVmaXhGKC9GX3BGO0ZgcC9GY3BRLnRoaW5tYXRoc3BhY2VGKC9GZnBGZnMvRmhwRjtGaXBGW3FGXnFGYXFGY3FGZXFGZ3FGaXFGW3JGXXItRiQ2J0YsLUYkNiVGLC1JJW1zdXBHRiU2JS1GJDYmLUYtNjlRImZGKEYwRjNGNkY5RjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRmhuRltvLUZobzYzUTAmQXBwbHlGdW5jdGlvbjtGKEZbcEZecEZgcEZicEZlcEZncEZpcEZbcUZecUZhcUZjcUZlcUZncUZpcUZbckZdci1GJDYlRl9zLUYkNiMtRi02OVEieEYoRjBGM0Y2RjlGPEY+RkBGQkZFRkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm5GaG5GW28tRmhvNjNRIilGKC9GXHBRKHBvc3RmaXhGKEZkc0ZgcEZlcy9GZnBRMnZlcnl0aGlubWF0aHNwYWNlRihGaHNGaXBGW3FGXnFGYXFGY3FGZXFGZ3FGaXFGW3JGXXJGLC1JI21uR0YlNjlRIjJGKEYwRjNGNkZkb0Y8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZlb0Zbby8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRihGLC1GaG82M1EoJm1pbnVzO0YoRltwRl5wRmBwL0ZjcFEwbWVkaXVtbWF0aHNwYWNlRigvRmZwRmF2RmdwRmlwRltxRl5xRmFxRmNxRmVxRmdxRmlxRltyRl1yLUYkNiVGLC1GXnQ2JS1GJDYmLUYtNjlRImdGKEYwRjNGNkY5RjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRmhuRltvRmV0Rmh0RixGZnVGanVGLEYsRl91RmdvRmFvRmdvRlx1RixGLDcjNiMvKiYlJkRlbHRhRyIiIiUiVkdGYXcqKiUjUGlHRmF3LCYqJCktJSJmRzYjJSJ4RyIiI0Zhd0ZhdyokKS0lImdHRmp3Rlx4RmF3ISIiRmF3RmB3RmF3Rlt4RmF3 . Now try to imagine the solid formed by revolving the entire region about the x -axis. the next Maple plot is a point plot of the solid. p1:= tubeplot([x,0,0],x=0..s,radius=f(x)): p2:=tubeplot([x,0,0],x=0..s,radius=g(x)): display({p1,p2},axes=normal,orientation=[-60,75],style=patch); The volume of this solid of revolution is equal to 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, where a and b are the x coordinates of the points of intersection of the two curves. The integral is calculated. Int(Pi*(f(x)^2-g(x)^2),x=0..s): %=value(%); Example 3: The following procedure will allow one to see the rotation of the region bounded by 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 around the x-axis. Execute the commands, click on the plot and then use the toolbar to animate it. To see the rotation of a different curve, put it in place of sin(t) in the spacecurve command. To change the axis of rotation, put i*Pi/20 in a different position in the rotate command. restart: with(plots):with(plottools): setoptions3d(axes=normal, thickness=2, color=blue, frames=40, tickmarks=[4,4,4]); p:=spacecurve([sin(t),t,0], t=0..Pi): b:=array(0..40): q:=array(0..40): for i from 0 to 40 do b[i]:=rotate(p,0,i*Pi/20,0): od: for k from 1 to 40 do q[k]:=display(seq(b[j], j=1..k)); od: display( seq(q[i], i=1..40), insequence=true); (Click on this plot and then the play arrow on the animation button bar.)

Consider the region bounded by the curve 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, between x=0, 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, and the x -axis. Plot this region. Plot the 3-dimensional shape using the tubeplot command. Then calculate the exact value of the volume of the solid formed by revolving the region about the x-axis. restart;with(plots):

Consider the region bounded by the curves NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYoLyUqbWF0aGNvbG9yR0ZELyUvbWF0aGJhY2tncm91bmRHRkQvJStmb250ZmFtaWx5R0YyLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGKC8lKW1hdGhzaXplR0Y1LUYkNidGLC1GJDYmLUYtNjlRImZGKEYwRjNGNkY5RjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRmhuRltvLUkjbW9HRiU2M1EwJkFwcGx5RnVuY3Rpb247RigvJSVmb3JtR1EmaW5maXhGKC8lJmZlbmNlR0Y4LyUqc2VwYXJhdG9yR0Y4LyUnbHNwYWNlR1EkMGVtRigvJSdyc3BhY2VHRmFwLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lMGZvbnRfc3R5bGVfbmFtZUdGVy8lJXNpemVHRjUvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGRC1GJDYlLUZlbzYzUSIoRigvRmlvUSdwcmVmaXhGKC9GXHBGO0ZdcC9GYHBRLnRoaW5tYXRoc3BhY2VGKC9GY3BGZXIvRmVwRjtGZnBGaHBGW3FGXnFGYHFGYnFGZHFGZnFGaHFGanEtRiQ2Iy1GLTY5USJ4RihGMEYzRjZGOUY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZobkZbby1GZW82M1EiKUYoL0Zpb1EocG9zdGZpeEYoRmNyRl1wRmRyL0ZjcFEydmVyeXRoaW5tYXRoc3BhY2VGKEZnckZmcEZocEZbcUZecUZgcUZicUZkcUZmcUZocUZqcUYsLUZlbzYzUSI9RihGaG9GW3BGXXAvRmBwUS90aGlja21hdGhzcGFjZUYoL0ZjcEZoc0ZkcEZmcEZocEZbcUZecUZgcUZicUZkcUZmcUZocUZqcS1GJDYoRiwtRiQ2JUYsLUklbXN1cEdGJTYlRmpyLUkjbW5HRiU2OVEiMkYoRjBGM0Y2L0Y6RjhGPEY+RkBGQkZFRkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm4vRmluUSdub3JtYWxGKEZbby8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRihGLC1GZW82M1EoJm1pbnVzO0YoRmhvRltwRl1wL0ZgcFEwbWVkaXVtbWF0aHNwYWNlRigvRmNwRl91RmRwRmZwRmhwRltxRl5xRmBxRmJxRmRxRmZxRmhxRmpxLUYkNiUtRmJ0NjlRIjZGKEYwRjNGNkZldEY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZmdEZbby1GZW82M1ExJkludmlzaWJsZVRpbWVzO0YoRmhvRltwRl1wRl9wRmJwRmRwRmZwRmhwRltxRl5xRmBxRmJxRmRxRmZxRmhxRmpxRmpyLUZlbzYzUSIrRihGaG9GW3BGXXBGXnVGYHVGZHBGZnBGaHBGW3FGXnFGYHFGYnFGZHFGZnFGaHFGanEtRmJ0NjlRIzEwRihGMEYzRjZGZXRGPEY+RkBGQkZFRkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm5GZnRGW29GLEYsNyM2Iy8tJSJmRzYjJSJ4RywoKiQpRmV2IiIjIiIiRmp2KiYiIidGanZGZXZGanYhIiIiIzVGanY= and 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. Plot this region. Plot the 3-dimensional shape using the tubeplot command. Then calculate the exact value of the volume of the solid formed by revolving the region about the x-axis. restart;with(plots):

Consider the region bounded by the curves 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 and 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. Plot this region. Plot the 3-dimensional shape (when this region is revolved aroundt he y-axis) using the tubeplot command. Then calculate the exact value of the volume of the solid formed by revolving the region about the y-axis.

Arc Length and Surface Area Name:

with(plots): a:=plot(sqrt(x),x=0..3,y=0..sqrt(3)): b:=polygonplot([[.8^2,.8],[2,.8],[2,sqrt(2)]]): c:=textplot([1.4,.72,convert([68],bytes)],font=[SYMBOL,12]): d:=textplot([1.55,.72,`x`]): e:=textplot([2.15,1.1,convert([68],bytes)],font=[SYMBOL,12]): f:=textplot([2.3,1.1,`y`]): g:=textplot([1.1,1.2,convert([68],bytes)],font=[SYMBOL,12]): h:=textplot([1.23,1.2,`L`],font=[COURIER,12]): display({a,b,c,d,e,f,g,h},thickness=2); 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 Therefore: If f ' is continuous on [a, b], then the length of the curve 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 on the interval is given by 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 Alternately, if g ' is continuous on [c, d], then the length of the curve 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 on the interval is given by 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

Find the length of NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYoLyUqbWF0aGNvbG9yR0ZELyUvbWF0aGJhY2tncm91bmRHRkQvJStmb250ZmFtaWx5R0YyLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGKC8lKW1hdGhzaXplR0Y1LUYkNidGLC1GJDYmLUYtNjlRImZGKEYwRjNGNkY5RjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRmhuRltvLUkjbW9HRiU2M1EwJkFwcGx5RnVuY3Rpb247RigvJSVmb3JtR1EmaW5maXhGKC8lJmZlbmNlR0Y4LyUqc2VwYXJhdG9yR0Y4LyUnbHNwYWNlR1EkMGVtRigvJSdyc3BhY2VHRmFwLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lMGZvbnRfc3R5bGVfbmFtZUdGVy8lJXNpemVHRjUvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGRC1GJDYlLUZlbzYzUSIoRigvRmlvUSdwcmVmaXhGKC9GXHBGO0ZdcC9GYHBRLnRoaW5tYXRoc3BhY2VGKC9GY3BGZXIvRmVwRjtGZnBGaHBGW3FGXnFGYHFGYnFGZHFGZnFGaHFGanEtRiQ2Iy1GLTY5USJ4RihGMEYzRjZGOUY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZobkZbby1GZW82M1EiKUYoL0Zpb1EocG9zdGZpeEYoRmNyRl1wRmRyL0ZjcFEydmVyeXRoaW5tYXRoc3BhY2VGKEZnckZmcEZocEZbcUZecUZgcUZicUZkcUZmcUZocUZqcUYsLUZlbzYzUSI9RihGaG9GW3BGXXAvRmBwUS90aGlja21hdGhzcGFjZUYoL0ZjcEZoc0ZkcEZmcEZocEZbcUZecUZgcUZicUZkcUZmcUZocUZqcS1GJDYmLUYtNjlRJGNvc0YoRjBGM0Y2L0Y6RjhGPEY+RkBGQkZFRkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm4vRmluUSdub3JtYWxGKEZbb0Zkb0ZcckYsRixGLDcjNiMvLSUiZkc2IyUieEctJSRjb3NHRmd0 on the interval 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. with(plots): f:=x->cos(x); animatecurve(f(x),x=0..Pi,color=blue,thickness=2,frames=50,scaling=constrained); #Click on the plot and use the arrow above to animate it. Try to estimate the length of the curve using line segments and then use an integral to find the length. fx:=D(f); L:=int(sqrt(1+fx(x)^2),x=0..Pi); evalf(%);

Find the length of the curve 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 from x = 0 to x = 1. restart;with(plots): f:=x->x^3: animatecurve(f(x),x=0..1,color=green,thickness=2,scaling=constrained); fx:=D(f); L:=Int(sqrt(1+fx(x)^2),x=0..1); L1:=int(sqrt(1+fx(x)^2),x=0..1); evalf(L1); Does this answer seem reasonable? Could you evaluate the integral by hand?

The Gateway Arch in St. Louis is approximately modeled by the equation NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYoLyUqbWF0aGNvbG9yR0ZELyUvbWF0aGJhY2tncm91bmRHRkQvJStmb250ZmFtaWx5R0YyLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGKC8lKW1hdGhzaXplR0Y1LUYkNidGLC1GJDYmLUYtNjlRImZGKEYwRjNGNkY5RjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRmhuRltvLUkjbW9HRiU2M1EwJkFwcGx5RnVuY3Rpb247RigvJSVmb3JtR1EmaW5maXhGKC8lJmZlbmNlR0Y4LyUqc2VwYXJhdG9yR0Y4LyUnbHNwYWNlR1EkMGVtRigvJSdyc3BhY2VHRmFwLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lMGZvbnRfc3R5bGVfbmFtZUdGVy8lJXNpemVHRjUvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGRC1GJDYlLUZlbzYzUSIoRigvRmlvUSdwcmVmaXhGKC9GXHBGO0ZdcC9GYHBRLnRoaW5tYXRoc3BhY2VGKC9GY3BGZXIvRmVwRjtGZnBGaHBGW3FGXnFGYHFGYnFGZHFGZnFGaHFGanEtRiQ2Iy1GLTY5USJ4RihGMEYzRjZGOUY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZobkZbby1GZW82M1EiKUYoL0Zpb1EocG9zdGZpeEYoRmNyRl1wRmRyL0ZjcFEydmVyeXRoaW5tYXRoc3BhY2VGKEZnckZmcEZocEZbcUZecUZgcUZicUZkcUZmcUZocUZqcUYsLUZlbzYzUSI9RihGaG9GW3BGXXAvRmBwUS90aGlja21hdGhzcGFjZUYoL0ZjcEZoc0ZkcEZmcEZocEZbcUZecUZgcUZicUZkcUZmcUZocUZqcS1GJDYmLUkjbW5HRiU2OVEpNjkzLjg1OTdGKEYwRjNGNi9GOkY4RjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuL0ZpblEnbm9ybWFsRihGW28tRmVvNjNRKCZtaW51cztGKEZob0ZbcEZdcC9GYHBRMG1lZGl1bW1hdGhzcGFjZUYoL0ZjcEZndEZkcEZmcEZocEZbcUZecUZgcUZicUZkcUZmcUZocUZqcS1GJDYmLUZddDY5USg2OC43NjcyRihGMEYzRjZGYHRGPEY+RkBGQkZFRkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm5GYXRGW28tRmVvNjNRMSZJbnZpc2libGVUaW1lcztGKEZob0ZbcEZdcEZfcEZicEZkcEZmcEZocEZbcUZecUZgcUZicUZkcUZmcUZocUZqcS1GJDYmLUYtNjlRJWNvc2hGKEYwRjNGNkZgdEY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZhdEZbb0Zkby1GJDYlRl5yLUYkNiVGLC1GJDYlLUZddDY5USowLjAxMDAzMzNGKEYwRjNGNkZgdEY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZhdEZbb0ZedUZqckYsRl1zRixGLEYsRixGLDcjNiMvLSUiZkc2IyUieEcsJi0lJkZsb2F0RzYkIigoZlFwISIlIiIiKiYtRmh2NiQiJ3N3b0Zbd0Zcdy0lJWNvc2hHNiMqJi1GaHY2JCInTC41ISIoRlx3RmV2Rlx3Rlx3ISIi where f(x) gives the height of the arch in feet and x is the distance from directly below the peak to the base of the arch. Note: 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 . Maple understands the cosh function. Use the equation to find the height and length of the arch and its width at ground level. restart;

A cable hanging between two poles of the same height can be shown to form a catenary curve which can be modeled by 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 where a is the height above the ground of the lowest point of the cable, b is to be determined by the height of the poles (you must find b!!!), f(x) is the height of the cable and x is the distance from directly below the lowest point to either of the poles. Find the length of a cable hanging between 2 poles which are 40 meters apart if the lowest point of the cable is 14 meters above the ground and the poles are 15 meters high. restart;