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soldi70 [24.7K]
3 years ago
10

The equations that must be solved for maximum or minimum values of a differentiable function w=​f(x,y,z) subject to two constrai

nts ​g(x,y,z)=0 and ​h(x,y,z)=​0, where g and h are also​ differentiable, are gradientf=lambdagradientg+mugradient​h, ​g(x,y,z)=​0, and ​h(x,y,z)=​0, where lambda and mu ​(the Lagrange​ multipliers) are real numbers. Use this result to find the maximum and minimum values of ​f(x,y,z)=xsquared+ysquared+zsquared on the intersection between the cone zsquared=4xsquared+4ysquared and the plane 2x+4z=2.
Mathematics
1 answer:
sammy [17]3 years ago
5 0

The Lagrangian is

L(x,y,z,\lambda,\mu)=x^2+y^2+z^2+\lambda(4x^2+4y^2-z^2)+\mu(2x+4z-2)

with partial derivatives (set equal to 0)

L_x=2x+8\lambda x+2\mu=0\implies x(1+4\lambda)+\mu=0

L_y=2y+8\lambda y=0\implies y(1+4\lambda)=0

L_z=2z-2\lambda z+4\mu=0\implies z(1-\lambda)+2\mu=0

L_\lambda=4x^2+4y^2-z^2=0

L_\mu=2x+4z-2=0\implies x+2z=1

Case 1: If y=0, then

4x^2-z^2=0\implies4x^2=z^2\implies2|x|=|z|

Then

x+2z=1\implies x=1-2z\implies2|1-2z|=|z|\implies z=\dfrac25\text{ or }z=\dfrac23

\implies x=\dfrac15\text{ or }x=-\dfrac13

So we have two critical points, \left(\dfrac15,0,\dfrac25\right) and \left(-\dfrac13,0,\dfrac23\right)

Case 2: If \lambda=-\dfrac14, then in the first equation we get

x(1+4\lambda)+\mu=\mu=0

and from the third equation,

z(1-\lambda)+2\mu=\dfrac54z=0\implies z=0

Then

x+2z=1\implies x=1

4x^2+4y^2-z^2=0\implies1+y^2=0

but there are no real solutions for y, so this case yields no additional critical points.

So at the two critical points we've found, we get extreme values of

f\left(\dfrac15,0,\dfrac25\right)=\dfrac15 (min)

and

f\left(-\dfrac13,0,\dfrac23\right)=\dfrac59 (max)

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Refer to the diagram below. We need to find the areas of the green and blue regions, then subtract to get the area of the orange track only.

The larger green region is composed of a rectangle of dimensions 200 meters by 4+42+4 = 50 meters, along with two semicircles that combine to make a full circle. This circle has radius 25 meters.

The green rectangle has area 200*50 = 10000 square meters. The green semicircles combine to form an area of pi*r^2 = pi*25^2 = 625pi square meters. In total, the full green area is 10000+625pi square meters. I'm leaving things in terms of pi for now. The approximation will come later.

The blue area is the same story, but smaller dimensions. The blue rectangle has dimensions 200 meters by 42 meters, so its area is 200*42 = 8400 square meters. The blue semicircular pieces combine to a circle with area pi*r^2 = pi*21^2 = 441pi square meters. In total, the blue region has area 8400+441pi square meters.

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