1) Find the extreme values of f subject to both constraints. (If an answer does
ID: 2837803 • Letter: 1
Question
1)
Find the extreme values of f subject to both constraints. (If an answer does not exist, enter DNE.)
f(x, y, z) = 3x - y - 3z; x + y - z = 0, x2 + 2z2 = 3
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2)
Use Lagrange multipliers to find the shortest distance, d, from the point
(4, 0, ?5) to the plane x + y + z = 1.
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3)
The base of an aquarium with given volume V is made of slate and the sides are made of glass. If the slate costs five times as much (per unit area) as glass, use Lagrange multipliers to find the dimensions of the aquarium that minimize the cost of the materials. (Enter the dimensions as a comma separated list.)
Explanation / Answer
3.)
Cost C = 5pLW + 2pV/W + 2pV/L..........where p is price of glass
dC/dL = 5pW - 2pV/L^2
dC/dW = 5pL - 2pV/W^2
You need both of these to be zero. Can you see that these lead to L = W ?
Therefore 5L^3 = 2V ----> L = (2V/5)^(1/3)
1.)
Boundary extrema:
If g(x,y,z) = x + y - z and h(x,y,z) = x
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