An Oceanographic Vessel Suspends a Paraboloid-Shaped Net Below the Ocean $1200$

question 10

Multiple Choice

An oceanographic vessel suspends a paraboloid-shaped net below the ocean at depth of $1200$ feet, held open at the top by a circular metal ring of radius $20$ feet, with bottom $90$ feet below the ring and just touching the ocean floor.Set up coordinates with the origin at the point where the net touches the ocean floor and with z measured upward.

• A) \(\int _ { - 20 } ^ { 20 } \int _ { - \sqrt { 400 - y } ^ { 2 } } ^ { \sqrt { 400 - y ^ { 2 } } } \left( 2 x \left( \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { i } + \left( 1290 + x e ^ { - 3 x ^ { 2 } } \right) \vec { j } - \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \left( 1290 - \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { k } \right) \cdot \left( \frac { x } { 2 } \vec { i } + \frac { y } { 2 } \vec { j } - \vec { k } \right) d x d y\)
• B) \(\int _ { - 20 } ^ { 20 } \int _ { - y ^ { 400 - y ^ { 2 } } } ^ { \sqrt { 400 - y ^ { 2 } } } \left( 2 x \left( \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { i } + \left( 1290 + x e ^ { - 3 x ^ { 2 } } \right) \vec { j } - \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \left( 1290 - \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { k } \right) d x d y\)
• C) \(\int _ { - 20 } ^ { 20 } \int _ { - \sqrt { 400 - y } ^ { 2 } } ^ { \sqrt { 400 - y ^ { 2 } } } \left( 2 x \left( \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { i } - \left( 1290 + x e ^ { - 3 x ^ { 2 } } \right) \vec { j } + \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \left( 1290 - \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { k } \right) \cdot \left( \frac { x } { 2 } \vec { i } + \frac { y } { 2 } \vec { j } - \vec { k } \right) d x d y\)
• D) \(\int _ { - 20 } ^ { 20 } \int _ { - \sqrt { 20 - y ^ { 2 } } } ^ { \sqrt { 20 - y ^ { 2 } } } \left( 2 x \left( \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { i } - \left( 1290 + x e ^ { - 3 x ^ { 2 } } \right) \vec { j } + \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \left( 1290 - \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) \vec { k } \right) \cdot \left( \frac { x } { 2 } \vec { i } + \frac { y } { 2 } \vec { j } - \vec { k } \right) d x d y\)
• E) \(\int _ { - 20 } ^ { 20 } \int _ { - \sqrt { 400 - y ^ { 2 } } } ^ { \sqrt { 400 - y ^ { 2 } } } \left( 2 x \left( \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) - \left( 1290 + x e ^ { - 3 x ^ { 2 } } \right) + \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \left( 1290 - \frac { 9 } { 40 } \left( x ^ { 2 } + y ^ { 2 } \right) \right) d x d y \right.\)

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