#$&* course MTH 277 6/29 Question: `q001. If z = 1, then what is the resulting equation in y and z? Put this equation into the standard form of a conic section, identify that conic section, and sketch it.YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
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Given Solution: If z = 1 then the equation is x^2 / 25 + y^2 / 4 - 1 = 0, which can be written x^2 / 25 + y^2 / 4 = 1. This is of the form x^2 / a^2 + y^2 / b^2 = 1, with a = 5 and b = 2. This is an ellipse centered at the origin, with semi-major axis 5 in the x direction and semi-minor axis 2 in the y direction. Its graph looks like this: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: ********************************************* Question: `q002. Answer the same for z = 2. Compare your sketch to your sketch for the first question. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If z = 2 we get x^2 / 25 + y^2 / 4 - 2^2 = 0, or x^2 / 25 + y^2 / 4 = 4 which is put into standard form (which requires 1 on the right-hand side) by dividing both sides by 4. We obtain x^2 / 100 + y^2 / 16 = 1, which is an ellipse with semi-axes 10 and 4. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: If z = 2 we get x^2 / 25 + y^2 / 4 - 2^2 = 0, or x^2 / 25 + y^2 / 4 = 4 which is put into standard form (which requires 1 on the right-hand side) by dividing both sides by 4. We obtain x^2 / 100 + y^2 / 16 = 1, which is an ellipse with semi-axes 10 and 4. Its graph, and the graph of the ellipse from the preceding problem, looks like this: The first ellipse is at z = 1, which is the plane parallel to the xy plane, lying 1 unit above the xy plane. The second ellipse is at z = 2, which is the plane parallel to the xy plane, lying 2 units above the xy plane. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating:ok ********************************************* Question: `q003. Answer the same for z = 3, and make the same comparison. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: You will find that for z = 3 we get the ellipse x^2 / 225 + y^2 / 36 = 1 with semi-axes 15 and 6. A sketch of all three ellipses will show the ellipses growing linearly with the value of z, and you should visualize and attempt to sketch the ellipses on their respective planes z = 1, z = 2 and z = 3. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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Given Solution: You will find that for z = 3 we get the ellipse x^2 / 225 + y^2 / 36 = 1 with semi-axes 15 and 6. A sketch of all three ellipses will show the ellipses growing linearly with the value of z, and you should visualize and attempt to sketch the ellipses on their respective planes z = 1, z = 2 and z = 3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating:ok ********************************************* Question: `q004. If your sketches for the preceding three questions were made on transparent material and stacked, with their centers in a vertical line and the first being 1 unit above the tabletop, the second being 2 units above, and the third three units above, what 3-dimensional shape would they suggest? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: It would suggest a cone shape. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: ok ********************************************* Question: `q005. What would the intersection of this 3-dimensional shape with the x-z plane look like? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: it would be a bunch of ellipses on top of eachother. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The plot would be a 3-dimensional surface consisting of a series of 'stacked' ellipses. Plotted along with the plane z = 1 inside the rectangular region indicated below, in which x and y vary from -5 to 5 and z from 0 to 3, we see the elliptical z = 1 intersection. Adding the z = 2 plane to the figure, we see part of the ellipse as it intersects that plane: From the 'front' the figure looks like this: From the 'side' the lower part looks like this: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating:ok ********************************************* Question: `q006. What would the intersection of this 3-dimensional shape with the x-y plane look like? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: In the x-y plane our x and y values are both zero so our equation becomes jus -z^2 = 0 with solution z = 0. Thus when restricted to the x-y plane the surface consists of just the one point (0, 0, 0), the origin. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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Given Solution: In the x-y plane our x and y values are both zero so our equation becomes jus -z^2 = 0 with solution z = 0. Thus when restricted to the x-y plane the surface consists of just the one point (0, 0, 0), the origin. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating:ok ********************************************* Question: `q007. The plane x = 1 is parallel to the y-z plane, but passes through the x axis at x coordinate 1. What would the intersection of this plane with the surface look like? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If x = 1 the surface becomes 1^2 / 25 + y^2 / 4 - z^2 = 0 so that z^2 = y^2 / 4. Solving for z we get z = +- y / 2. The intersection of the surface with the plane x = 1 consists of the two straight lines z = y / 2 and z = -y / 2, both through the origin, one with slope 1/2 the other with slope -1/2. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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Given Solution: If x = 1 the surface becomes 1^2 / 25 + y^2 / 4 - z^2 = 0 so that z^2 = y^2 / 4. Solving for z we get z = +- y / 2. The intersection of the surface with the plane x = 1 consists of the two straight lines z = y / 2 and z = -y / 2, both through the origin, one with slope 1/2 the other with slope -1/2. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique rating:ok ********************************************* Question: `q008. If y = 2, then what is the resulting equation in x and z? Put this equation into the standard form of a conic section, identify that conic section and sketch it. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If y = 2 we get x^2 / 25 + 2^2 / 4 - z^2 = 0 so that x^2 / 25 - z^2 = -1 and -x^2 / 25 + z^2 = 1. This is a hyperbola with vertices at (0, 1) and (0, -1) in the x-z plane, asymptotic to the lines z = x / 5 and z = -x / 5 (the lines with slope 1/5 and -1/5, through the origin). confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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Given Solution: If y = 2 we get x^2 / 25 + 2^2 / 4 - z^2 = 0 so that x^2 / 25 - z^2 = -1 and -x^2 / 25 + z^2 = 1. This is a hyperbola with vertices at (0, 1) and (0, -1) in the x-z plane, asymptotic to the lines z = x / 5 and z = -x / 5 (the lines with slope 1/5 and -1/5, through the origin). The figure below shows the upper half of the hyperbola: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating:ok ********************************************* Question: `q009. Repeat the above for y = 4, then for y = 6. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If y = 4 we get x^2 / 25 - z^2 = -4 which rearranges to standard form z^2 / 4 - x^2 / 100 = 1. The vertices are at (0, 2) and (0, -2), and the asymptotes are still the lines z = 1/5 x and z -1/5 x. For y = 6 the asymptotes are still the same, with vertices (0, 3) and (0, -3). confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: If y = 4 we get x^2 / 25 - z^2 = -4 which rearranges to standard form z^2 / 4 - x^2 / 100 = 1. The vertices are at (0, 2) and (0, -2), and the asymptotes are still the lines z = 1/5 x and z -1/5 x. For y = 6 the asymptotes are still the same, with vertices (0, 3) and (0, -3). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating:ok ********************************************* Question: `q010. Your last three sketches describe the intersection of the surface x^2 / 25 + y^2 / 4 - z^2 = 0 with the planes y = 2, y = 4 and y = 6, each plane being parallel to the x-z plane and passing through the y axis at the indicated coordinate. Explain how your sketches are consistent with the surface as you described it, based on the three stacked graphs. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: All three are ellipses stacked confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 3
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