Answer:
- x + y ≤ 20,000
- y ≤ 5,000
- x ≥ 2y
Step-by-step explanation:
Given:
Total amount = $20,000
Assume
Government bonds = x
Municipal bonds = y
Computation:
1. Invest total money
x + y ≤ 20,000
2. invest no more than $5,000 into municipal bonds
y ≤ 5,000
3. at least twice as much into government bonds.
x ≥ 2y
Answer: 3.3
Steps: 1) 1+1+2+3+3+4+4+5+5+5=33
2) 33/10=3.3
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