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Marat540 [252]
3 years ago
8

EXAMPLE 5 Find the maximum value of the function f(x, y, z) = x + 2y + 9z on the curve of intersection of the plane x − y + z =

1 and the cylinder x2 + y2 = 1. SOLUTION We maximize the function f(x, y, z) = x + 2y + 9z subject to the constraints g(x, y, z) = x − y + z = 1 and h(x, y, z) = x2 + y2 = 1. The Lagrange condition is ∇f = λ∇g + μ∇h, so we solve the equations
Mathematics
1 answer:
geniusboy [140]3 years ago
8 0

The Lagrangian,

L(x,y,z,\lambda,\mu)=x+2y+9z-\lambda(x-y+z-1)-\mu(x^2+y^2-1)

has critical points where its partial derivatives vanish:

L_x=1-\lambda-2\mu x=0

L_y=2+\lambda-2\mu y=0

L_z=9-\lambda=0

L_\lambda=x-y+z-1=0

L_\mu=x^2+y^2-1=0

L_z=0 tells us \lambda=9, so that

L_x=0\implies-8-2\mu x=0\implies x=-\dfrac4\mu

L_y=0\implies11-2\mu y=0\implies y=\dfrac{11}{2\mu}

Then with L_\mu=0, we get

x^2+y^2=\dfrac{16}{\mu^2}+\dfrac{121}{4\mu^2}=1\implies\mu=\pm\dfrac{\sqrt{185}}2

and L_\lambda=0 tells us

x-y+z=-\dfrac4\mu-\dfrac{11}{2\mu}+z=1\implies z=1+\dfrac{19}{2\mu}

Then there are two critical points, \left(\pm\frac8{\sqrt{185}},\mp\frac{11}{\sqrt{185}},1\pm\frac{19}{\sqrt{185}}\right). The critical point with the negative x-coordinates gives the maximum value, 9+\sqrt{185}.

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Which of the following could be the equation of the graph shown below? Check all that apply.
Mashutka [201]

Answer:

y = -2x + 5

Х-y= 9

2x + y = 4

y = 5

lmk if incorrect, please give brainliest

8 0
3 years ago
Find the volume of a pyramid with a square base, where the perimeter of the base is
grigory [225]

Answer:

90.3125 cm³

Step-by-step explanation:

Since the perimeter of the bottom square is 17, you can divide by four to find out what the lenght of each side is.

17/4=4.25 cm

We know the height of pyramid is 15 cm

The formula for the volume of a square pyramid is

V=a²(h/3)

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8 0
3 years ago
What is the product (2x-1)(x+4)
Ostrovityanka [42]
2x(x+4)-1(x+4)
2x^2+8x-x-4
2x^2+7x-4
8 0
3 years ago
Read 2 more answers
I WILL GIVE YOU 20 POINTS AND THE BRAINLYEST
fiasKO [112]

Answer/Step-by-step explanation:

✍️Slope of the line using two points, (2, 2) and (6, 10),

slope (m) = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 2}{6 - 2} = \frac{8}{4} = 2

✍️To find the equation of the line in slope-intercept form, we need to find the y-intercept (b).

Substitute x = 2, y = 2, and m = 2 in y = mx + b, and solve for b.

2 = (2)(2) + b

2 = 4 + b

2 - 4 = b

-2 = b

b = -2

Substitute m = 2 and b = -2 in y = mx + b.

✅The equation would be:

y = 2x + (-2)

y = 2x - 2

✍️To find the value of a, plug in (a, 8) as (x, y) into the equation of the line.

8 = 2(a) - 2

8 = 2a - 2

Add 2 to both sides

8 + 2 = 2a

10 = 2a

Divide both sides by 2

\frac{10}{2} = a

5 = a

a = 5

✍️To find the value of b, plug in (4, b) as (x, y) into the equation of the line.

b = 2(4) - 2

b = 8 - 2

b = 6

5 0
3 years ago
How do you solve this equation ( ASAP PLS )
makvit [3.9K]
<span>Solve this by substitution:

</span>Solve 5x - 4y = -13 for x:<span><span><span><span>5x </span>− <span>4y </span></span>+ <span>4y </span></span>= -<span>13 + <span>4y     </span></span></span>(Add 4y to both sides)
<span><span><u>5x  </u></span>= <span><span><u>4y - </u><u>13 </u></span><u> </u>  </span></span>(Divide both sides by 5)
 5          5

<span>x = <span><span><span><u>4</u></span> y </span>+ -<span><u>13</u><span>
</span></span></span></span>      5        5
You can use https://www.mathpapa.com/algebra-calculator.html
5x - 4y = -13, 8x +10y = 12 (enter both equations with a comma between them and it will solve for you and show you step by step.


8 0
3 years ago
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