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ValentinkaMS [17]
3 years ago
9

Pictures, x

Mathematics
1 answer:
masha68 [24]3 years ago
3 0
Wat fraction post a pic of the problem
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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
What is an equation of the line that passes through the points (5, 2) and (-5, -6)?​
lawyer [7]

Answer:

y=4/5x + 2

Step-by-step explanation:

y=mx+b

find slope which is 8/10=4/5

choose a y and plug into problem

8 0
2 years ago
Read 2 more answers
F(x)=4x <br> 2<br> +2x+6 How many distinct real number zeros does f have?
hoa [83]

Answer:

It has 0 distinct real number zeros. The function does not touch the x-axis

6 0
3 years ago
Read 2 more answers
What do they call the international, hula hoops championship?<br> Its for a math worksheet
lapo4ka [179]
They call it the Whirl Series ^.^
7 0
3 years ago
The yield of strawberry plants depends on the amount of fertilizer fed to the plants. Agricultural research shows that an acre o
andre [41]

Answer:

657 pounds

Step-by-step explanation:

Given

Represent the amount of fertilizer with x and the yield with y.

So, we have:

(x_1,y_1) = (70,630)

(x_2,y_2) = (100,900)

Required:

Determine the yield (y) when fertilizer (x) is 73ft^3

Using linear interpolation, we have:

y = y_1 + (x - x_1)\frac{(y_2 - y_1)}{(x_2 - x_1)}

Substitute the x and y values using

(x_1,y_1) = (70,630) and (x_2,y_2) = (100,900);

We have:

y = y_1 + (x - x_1)\frac{(y_2 - y_1)}{(x_2 - x_1)}

y = 630 + (x - 70)\frac{(900 - 630)}{(100 - 70)}

y = 630 + (x - 70)\frac{270}{30}

y = 630 + (x - 70)*9

Open bracket

y = 630 + 9x - 630

y = 9x - 630+630

y = 9x

To solve for y when x = 73.

We simply substitute 73 for x

y = 9x

y = 9 * 73

y = 657

<em>Hence, the yield for 73 cubic feet is 657 pounds</em>

6 0
3 years ago
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