Answer both Questions. Show all work. Focus on the Algebra. For #24, the answer

Question

Answer both Questions. Show all work. Focus on the Algebra.

For #24, the answer is 10.18. Just show the work.

Use Lagrange multipliers to find the minimum distance from the circle (x-2)2 + y2 = 1 to the point (0,11) Ans. 10.18 A cargo container (in the shape of a rectangular solid) must have a volume of 580 cubic feet. The bottom will cost $6 per square foot to construct and the sides and the top will cost $4 per square foot to construct. Use Lagrange multipliers to find the dimensions of the container of this volume that has minimum cost.

Explanation / Answer

Let Q(x,y,z) be a point on the circle and P(0,11) be the given point.

Then (x-2)^2+y^2=1 and the distance PQ is given by: (PQ)^2 = (x)^2+(y-11)^2

So in effect we require a minimum for f(x,y) = (x)^2+(y-11)^2

with the subsidiary condition that: (x-2)^2+y^2=1

g= (x-2)^2+y^2 - 1

f= (x)^2+(y-11)^2

Write f ? = f+?g where ? is a Lagrange multiplier. Then: f?? = 0 is: 2(x) + ?( 2(x-2)) = 0 .......... (1)

hence x+?(x-2) =0 => x( 1+ ?) - 2 ? =0 => x= 2? / (1+?)

fy? = 0 is 2(y-11) + ? (2y)= 0 ........... (2)

hence y + ?y -11 =0 => y = 11/(1+?)

Also (x-2)^2+y^2=1  ....................... (4)

Substitute in (4) for x,y and we get:

(2? / (1+?)-2)^2+ (11/(1+?) )^2 =1

(-2/(1+?))^2 + (11/(1+?) )^2 =1

4+121 = (1+?)^2

125 = (1+?)^2

1+? =11.18

? = 10.18

so, we have x = 2*10.18 / (1+10.18) = 1.8211

and y= 11/(1+? ) = 11/(1+4) = 0.9839

so the distance is: sqrt( (x)^2+(y-11)^2) = sqrt(103.64) = 10.18

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