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3 years ago

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Please explain and solve page, solve and explain for 56 points, thanks :D Plz help me :)Heeeeeeeellpppp my test is tomorrow and

i need to know this!

1 answer:

TEA [102]3 years ago

8 0

10> Geometric
11:geometric
12:arothmetic

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The plane x + y + z = 12 intersects paraboloid z = x^2 + y^2 in an ellipse.(a) Find the highest and the lowest points on the ell

emmasim [6.3K]

Answer:

a)

Highest (-3,-3)

Lowest (2,2)

b)

Farthest (-3,-3)

Closest (2,2)

Step-by-step explanation:

To solve this problem we will be using Lagrange multipliers.

a)

Let us find out first the restriction, which is the projection of the intersection on the XY-plane.

From x+y+z=12 we get z=12-x-y and replace this in the equation of the paraboloid:

$\bf 12-x-y=x^2+y^2\Rightarrow x^2+y^2+x+y=12$

completing the squares:

$\bf x^2+y^2+x+y=12\Rightarrow (x+1/2)^2-1/4+(y+1/2)^2-1/4=12\Rightarrow\\\\\Rightarrow (x+1/2)^2+(y+1/2)^2=12+1/2\Rightarrow (x+1/2)^2+(y+1/2)^2=25/2$

and we want the maximum and minimum of the paraboloid when (x,y) varies on the circumference we just found. That is, we want the maximum and minimum of

$\bf f(x,y)=x^2+y^2$

subject to the constraint

$\bf g(x,y)=(x+1/2)^2+(y+1/2)^2-25/2=0$

Now we have

$\bf \nabla f=(\displaystyle\frac{\partial f}{\partial x},\displaystyle\frac{\partial f}{\partial y})=(2x,2y)\\\\\nabla g=(\displaystyle\frac{\partial g}{\partial x},\displaystyle\frac{\partial g}{\partial y})=(2x+1,2y+1)$

Let $\bf \lambda$ be the Lagrange multiplier.

The maximum and minimum must occur at points where

$\bf \nabla f=\lambda\nabla g$

that is,

$\bf (2x,2y)=\lambda(2x+1,2y+1)\Rightarrow 2x=\lambda (2x+1)\;,2y=\lambda (2y+1)$

we can assume (x,y)≠ (-1/2, -1/2) since that point is not in the restriction, so

$\bf \lambda=\displaystyle\frac{2x}{(2x+1)} \;,\lambda=\displaystyle\frac{2y}{(2y+1)}\Rightarrow \displaystyle\frac{2x}{(2x+1)}=\displaystyle\frac{2y}{(2y+1)}\Rightarrow\\\\\Rightarrow 2x(2y+1)=2y(2x+1)\Rightarrow 4xy+2x=4xy+2y\Rightarrow\\\\\Rightarrow x=y$

Replacing in the constraint

$\bf (x+1/2)^2+(x+1/2)^2-25/2=0\Rightarrow (x+1/2)^2=25/4\Rightarrow\\\\\Rightarrow |x+1/2|=5/2$

from this we get

<em>x=-1/2 + 5/2 = 2 or x = -1/2 - 5/2 = -3 </em>

<em> </em>

and the candidates for maximum and minimum are (2,2) and (-3,-3).

Replacing these values in f, we see that

f(-3,-3) = 9+9 = 18 is the maximum and

f(2,2) = 4+4 = 8 is the minimum

b)

Since the square of the distance from any given point (x,y) on the paraboloid to (0,0) is f(x,y) itself, the maximum and minimum of the distance are reached at the points we just found.

We have then,

(-3,-3) is the farthest from the origin

(2,2) is the closest to the origin.

3 0

3 years ago

11c - 2d = -2 <br> c + 8d = 8 <br> using the substitution method

Darya [45]

<span>A) 11c - 2d = -2
B) c + 8d = 8
</span><span>B) c = 8 - 8d then substitute this into A)

</span><span>A) 88 -88d - 2d = -2
A) 90 = 90d
d = 1
c = 0

</span>

4 0

3 years ago

Tiana can clear a football field of debris in 3 hours. Jacob can clear the same field in 2 hours. If they decide to clear the fi

Marianna [84]

You must find the relative rates of both workers...

T=f/3 and J=f/2 so if they work together to clear the field then:

ft/3+ft/2=f make all terms have a common denominator of 6

(2/2)(ft/3)+(3/3)(ft/2)=(6/6)f

2ft/6+3ft/6=6f/6 multiply both sides by 6

2ft+3ft=6f divide both sides by f

2t+3t=6 combine like terms on left side

5t=6 divide both sides by 5

t=6/5 hr

t=1 1/5 hr

t=1 hr 12 min

8 0

3 years ago

£730 is divided between June, Sushil & Fiona so that June gets twice as much as Sushil, and Sushil gets three times as much

olga2289 [7]

Answer:

idk mush

Step-by-step explanation:

i dont have a clue

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3 years ago

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I need help! <br><br> What is 5+5?

-Dominant- [34]

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the answer is 10!!!!!!

good luck

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3 years ago

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