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The drawing on the right shows the plan and elevation of a right cone - Leaving Cert DCG - Question A-4 - 2009
Question A-4
The drawing on the right shows the plan and elevation of a right cone. A 3D graphic is also given below. A point P on the curved surface is shown in the plan. (a) L... show full transcript
Worked Solution & Example Answer:The drawing on the right shows the plan and elevation of a right cone - Leaving Cert DCG - Question A-4 - 2009
Step 1
Locate point P in elevation and draw the projections of a sphere which rests on the horizontal plane and which touches the cone at point P.
Answer
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Locating Point P in Elevation:
- Start by identifying point P's coordinates, given in the plan.
- From the plan, transfer point P vertically to the elevation view, marking it at (2,1) on the Y-axis.
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Plan of Required Sphere:
- The radius of the sphere can be determined by its contact with the cone at point P, ensuring that it rests on the horizontal plane.
- Calculate the maximum radius using the cone's dimensions which will be shown at edge of elevation in (1,1).
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Elevation of Sphere (2,2):
- Draw the outline of the sphere in elevation, marking its center at coordinates (2,2) to denote the position of contact. Make sure to maintain symmetry around the vertical centerline.
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Final Projection Drawings:
- Include the plan view of the sphere at (2,1) in relation to the cone in the main graphic to show how it touches the cone.
Step 2
Determine the traces of a plane which is tangential to the cone and the sphere as shown in the 3D graphic.
Answer
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Elevation and Plan of Circumscribing Cone:
- Begin by identifying points on the cone where the tangential plane will touch, noting the slopes. This can be visualized on the elevation as the apex and base of the cone.
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Horizontal Trace:
- Draw a horizontal line representing the trace on the horizontal plane at the required point of tangency between the cone and sphere.
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Vertical Trace:
- Similarly, draw the vertical trace through the center of elevation, ensuring it touches both the sphere and cone at their respective tangential points.