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NNADVOKAT [17]
3 years ago
13

I don't understand this question, what exactly is it asking? How do I solve it?

Mathematics
1 answer:
goldenfox [79]3 years ago
7 0
This question is saying that there is some point on the line where the x-value equals 4. So when x=4, what does y equal. I think this should be enough for you to be able to answer it now. If you need more help, though, just let me know
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The plane x+y+2z=8 intersects the paraboloid z=x2+y2 in an ellipse. Find the points on this ellipse that are nearest to and fart
DiKsa [7]

Answer:

The minimum distance of   √((195-19√33)/8)  occurs at  ((-1+√33)/4; (-1+√33)/4; (17-√33)/4)  and the maximum distance of  √((195+19√33)/8)  occurs at (-(1+√33)/4; - (1+√33)/4; (17+√33)/4)

Step-by-step explanation:

Here, the two constraints are

g (x, y, z) = x + y + 2z − 8  

and  

h (x, y, z) = x ² + y² − z.

Any critical  point that we find during the Lagrange multiplier process will satisfy both of these constraints, so we  actually don’t need to find an explicit equation for the ellipse that is their intersection.

Suppose that (x, y, z) is any point that satisfies both of the constraints (and hence is on the ellipse.)

Then the distance from (x, y, z) to the origin is given by

√((x − 0)² + (y − 0)² + (z − 0)² ).

This expression (and its partial derivatives) would be cumbersome to work with, so we will find the the extrema  of the square of the distance. Thus, our objective function is

f(x, y, z) = x ² + y ² + z ²

and

∇f = (2x, 2y, 2z )

λ∇g = (λ, λ, 2λ)

µ∇h = (2µx, 2µy, −µ)

Thus the system we need to solve for (x, y, z) is

                           2x = λ + 2µx                         (1)

                           2y = λ + 2µy                       (2)

                           2z = 2λ − µ                          (3)

                           x + y + 2z = 8                      (4)

                           x ² + y ² − z = 0                     (5)

Subtracting (2) from (1) and factoring gives

                     2 (x − y) = 2µ (x − y)

so µ = 1  whenever x ≠ y. Substituting µ = 1 into (1) gives us λ = 0 and substituting µ = 1 and λ = 0  into (3) gives us  2z = −1  and thus z = − 1 /2 . Subtituting z = − 1 /2  into (4) and (5) gives us

                            x + y − 9 = 0

                         x ² + y ² +  1 /2  = 0

however, x ² + y ² +  1 /2  = 0  has no solution. Thus we must have x = y.

Since we now know x = y, (4) and (5) become

2x + 2z = 8

2x  ² − z = 0

so

z = 4 − x

z = 2x²

Combining these together gives us  2x²  = 4 − x , so

2x²  + x − 4 = 0 which has solutions

x =  (-1+√33)/4

and

x = -(1+√33)/4.

Further substitution yeilds the critical points  

((-1+√33)/4; (-1+√33)/4; (17-√33)/4)   and

(-(1+√33)/4; - (1+√33)/4; (17+√33)/4).

Substituting these into our  objective function gives us

f((-1+√33)/4; (-1+√33)/4; (17-√33)/4) = (195-19√33)/8

f(-(1+√33)/4; - (1+√33)/4; (17+√33)/4) = (195+19√33)/8

Thus minimum distance of   √((195-19√33)/8)  occurs at  ((-1+√33)/4; (-1+√33)/4; (17-√33)/4)  and the maximum distance of  √((195+19√33)/8)  occurs at (-(1+√33)/4; - (1+√33)/4; (17+√33)/4)

4 0
3 years ago
Find the missing side of the triangle and leave the answer in simplest radical form.
tatuchka [14]

Answer:

x = \sqrt5

Step-by-step explanation:

Using Pythagoras theorem,

Square of longer side = Sum of square of other sides .

Therefore,

(\sqrt{11})^2 = x^2 + (\sqrt{6})^2\\\\11 = x^2 + 6\\\\11 - 6  = x^2\\\\5 = x^2\\\\x = \sqrt 5

6 0
3 years ago
Read 2 more answers
If two angles in a triangle measure 35 degrees and 47 degrees, what is the value of the third angle?
Furkat [3]

Answer:

278

Step-by-step explanation:360-82=278

8 0
2 years ago
Read 2 more answers
in the central city grand prix out of 30 cars that started the race,12 of them finished. what percent of the cars finished the r
olga_2 [115]
To get the answer we can use proportion.
30 ------------- 100%
12 --------------x
crossmultiply now
30x=12*100%
30x=1200%      /:30 (divide both sides by 30)
x=40% - its the percentage of cars which finished the race.
100%-40%=60% - its the answer
3 0
3 years ago
Read 2 more answers
3 cm<br> 4 cm<br> Find the area.<br> A = 1/2 (a*P)
ss7ja [257]
The answer would be 3:33333
3 0
2 years ago
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