Find a Vector Function That Represents the Curve of Intersection $x ^ { 2 } + 5 y ^ { 2 } + 5 z ^ { 2 } = 25$

question 82

Multiple Choice

Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid $x ^ { 2 } + 5 y ^ { 2 } + 5 z ^ { 2 } = 25$ and the parabolic cylinder $y = x ^ { 2 }$ .

• A) \(r ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + \sqrt { \frac { 25 - t ^ { 2 } - 5 t } { 5 } } \mathbf { k }\)
• B) \(\mathbf { r } ( t ) = t \mathbf { i } - t ^ { 2 } \mathbf { j } - \sqrt { \frac { 25 - t ^ { 2 } - 5 t } { 5 } } \mathbf { k }\)
• C) \(\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + \sqrt { \frac { 5 - t ^ { 2 } - 5 t ^ { 4 } } { 5 } } \mathbf { k }\)
• D) \(\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 4 } \mathbf { j } + \sqrt { \frac { 25 - t ^ { 2 } + 5 t } { 5 } } \mathbf { k }\)
• E) \(\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + \sqrt { \frac { 25 + t ^ { 2 } - 5 t ^ { 4 } } { 5 } } \mathbf { k }\)

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Definitions:

Rib Vaulting

Rib Vaulting is an architectural feature used in Gothic cathedrals, characterized by intersecting arched ribs that support a vaulted ceiling, enabling the construction of higher and more complex structures.

Saint-Étienne

Referring typically to the Cathedral of Saint-Étienne, which can relate to various churches named after Saint Stephen, known for their historical and architectural value.

Romanesque Churches

Churches built in a style prevalent in Europe during the 11th and 12th centuries, characterized by semi-circular arches, thick walls, and decorative carvings.

Bernard Of Clairvaux

A French abbot and the primary reformer of the Cistercian order in the 12th century, known for his contributions to Christian theology and mysticism.