course MTH 272 05/09,1830 assignment #023023.
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15:48:59 Query problem 7.2.6 intercepts and sketch graph of 2x - y + z = 4.
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RESPONSE --> 2x - y + z = 4 x-intercept: 2x - 0 + 0= 4 2x=4 x=2 (2,0,0) y-intercept: 2(0) - y + 0 = 4 -y=4 y=-4 (0,-4,0) z-intercept: 2(0) - 0 + z = 4 z=4 (0,0,4) confidence rating #$&* 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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15:49:13 List the intercepts of the graph.
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RESPONSE --> 2x - y + z = 4 x-intercept: 2x - 0 + 0= 4 2x=4 x=2 (2,0,0) y-intercept: 2(0) - y + 0 = 4 -y=4 y=-4 (0,-4,0) z-intercept: 2(0) - 0 + z = 4 z=4 (0,0,4) confidence rating #$&* 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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15:51:09 Describe the graph of the plane.
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RESPONSE --> The graph of the plane shows a triangle with vertices at each of the given intercepts (x,y, and z). confidence rating #$&* 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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15:57:49 If you released a marble on the plane at the point where it intercepts the z axis, it would roll down the incline. When the marble reached the xy plane would it be closer to the x axis or to the y axis?
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RESPONSE --> On the xy-plane, the z-value for both of the coordinates is 0, therefore both of these points lie directly on the plane. Assuming the marble released at the z-intercept rolls down the plane in the center, it should be equidistant from the x-axis and the y-axis when it reaches the xy-plane. (As an approximation, at point (1, 3, 0), midpoint of the line connecting the x and y intercepts.) confidence rating #$&* 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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15:59:56 If you were climbing the plane straight from your starting point to the point for the plane intercepts the z axis, with your climb be steeper if you started from the x intercept or from the y intercept?
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RESPONSE --> By comparing the apparent slope (rise/run) starting from the x-intercept and y-intercept and heading to the z-intercept, the xz line is steeper than yz. confidence rating #$&* 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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16:01:24 Query problem 7.2.34 (was 7.2.30) match y^2 = 4x^2 + 9z^2 with graph
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RESPONSE --> y^2 = 4x^2 + 9z^2 4x^2 - y^2 + 9z^2=0 (standard form) Graph B of the Elliptic Cone matches the standard form of this equation. confidence rating #$&* 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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16:01:38 Which graph matches the equation?
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RESPONSE --> Graph B, Elliptic Cone confidence rating #$&* 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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16:05:36 The graph couldn't be (e). Explain why not.
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RESPONSE --> When placed in standard form, the given equation is equal to zero (4x^2 - y^2 + 9z^2=0). Graph E is that of a Hyperboloid of Two Sheets which has a standard equation that is equal to one. The standard equation of a Hyperboloid of Two Sheets also involves subtracting two variables (i.e., two negative variables, one positive) as compared to the standard equation of the Elliptic Cone in which only one variable is subtracted (i.e., one negative variable, two positive). confidence rating #$&* 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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16:09:21 The graph could not be (c) because the picture shows that the surface is not defined for | y | < 1, while 4x^2 + 9z^2 = .25, for example, is the trace for y = 1/2, and is a perfectly good ellipse. State this in your own words.
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RESPONSE --> The standard equation for the Ellipsoid (Graph C) is set equal to one, whereas the standard equation of an Elliptic Cone (Graph B) is set equal to zero. This is why that at a value of 1/2 the ellipsoid is undefined but the elliptic cone is not. confidence rating #$&* 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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16:11:41 The graph couldn't be (c). Explain why not.
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RESPONSE --> Graph C, an ellipsoid, has a standard equation in which all of the variables are being added (i.e., three positive variables) and equal one. The standard equation we are given equals zero and has one variable being subtracted. This rules out an ellipsoid as the potential graph. confidence rating #$&* 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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16:24:07 The trace of this graph exists in each of the coordinate planes, and is an ellipse in each. The graph of the given equation consists only of a single point in the xz plane, since there y = 0 and 4x^2 + 9z^2 = 0 only if x = z = 0. Explain why the xy trace is not an ellipse.
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RESPONSE --> Since there is only one point in the xz-plane at (0,0,0), the trace in the xy-plane would come to this point as a parabola. If the trace here were an ellipse, there would be more points located in the xz-plane to produce the ellipse. Instead, this point in the xz-plane is the where the pointed heads of the two cones meet. confidence rating #$&* 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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16:27:16 What is the shape of the trace of the graph in the plane y = 1?
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RESPONSE --> At y=1: 4x^2 + 9z^2=1 The trace here is an ellipse as indicated by the formula. confidence rating #$&* 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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16:35:45 What is the shape of the trace of the graph in the plane x = 1?
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RESPONSE --> x=1: 4x^2 =y^2 - 9z^2 4=y^2 - 9z^2 Hyperbola confidence rating #$&* 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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16:39:05 What is the shape of the trace of the graph in the plane z = 1?
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RESPONSE --> 4x^2 - y^2 + 9z^2 = 0 9z^2=y^2 - 4x^2 9=y^2 - 4x^2 Hyperbola confidence rating #$&* 2" ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^