We can determine the mass of liquid mercury inside the vial by taking the diffrence between the filled vial and the empty vial. The weight of liquid mercury is 130.24 grams. The equivalent volume is obtained through the density. The volume is 9.626 cm3. If the substance inside is water, the volume is 130.632 grams
3 rolls of steamer + 2 packages of balloons = $10 -------------- (1)
2 rolls of steamer + 1 package of balloons = $6.25 --------- (2)
(2) x 2:
4 rolls of steamer + 2 package of balloons = $12.50 --------- (2a)
(2a) - (1):
1 roll of steamer = $2.50 ----------- Sub into (1)
3 ($2.50) + 2 packages of balloons = $10
$7.50 + 2 packages of balloons = $10
2 packages of balloons = $10 - $7.50
2 packages of balloons = $2.50
1 packages of balloons = $1.25
1 packages of balloons = $1.25
1 roll of steamer = $2.50
1 packages of balloons + 1 roll of steamer = $1.25 + $2.50 = $3.75
The tangent to through (1, 1, 1) must be perpendicular to the normal vectors to the surfaces and at that point.
Let . Then is the level curve . Recall that the gradient vector is perpendicular to level curves; we have
so that the gradient of at (1, 1, 1) is
For the surface , we have
so that . We can obtain a vector normal to by taking the cross product of the partial derivatives of , and evaluating that product for :
Now take the cross product of the two normal vectors to and :
The direction of vector (24, 8, -8) is the direction of the tangent line to at (1, 1, 1). We can capture all points on the line containing this vector by scaling it by . Then adding (1, 1, 1) shifts this line to the point of tangency on . So the tangent line has equation