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2 years ago

HELP WITH PROBABILITY! WILL MARK BRAINLIEST!!

Mathematics

1 answer:

IRISSAK [1]2 years ago

3 0

Answer:

4th and 3rd of August and then we will have a WONDERFUL Heathers day trip and I am going back home for the weekend of you and the kids

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Find the maximum and minimum values attained by f(x, y, z) = 5xyz on the unit ball x2 + y2 + z2 ≤ 1.

Allushta [10]

Check for critical points within the unit ball by solving for when the first-order partial derivatives vanish:
$f_x=5yz=0\implies y=0\text{ or }z=0$
$f_y=5xz=0\implies x=0\text{ or }z=0$
$f_z=5xy=0\implies x=0\text{ or }y=0$

Taken together, we find that (0, 0, 0) appears to be the only critical point on $f$ within the ball. At this point, we have $f(0,0,0)=0$ .

Now let's use the method of Lagrange multipliers to look for critical points on the boundary. We have the Lagrangian

$L(x,y,z,\lambda)=5xyz+\lambda(x^2+y^2+z^2-1)$

with partial derivatives (set to 0)

$L_x=5yz+2\lambda x=0$
$L_y=5xz+2\lambda y=0$
$L_z=5xy+2\lambda z=0$
$L_\lambda=x^2+y^2+z^2-1=0$

We then observe that

$xL_x+yL_y+zL_z=0\implies15xyz+2\lambda=0\implies\lambda=-\dfrac{15xyz}2$

So, ignoring the critical point we've already found at (0, 0, 0),

$5yz+2\left(-\dfrac{15xyz}2\right)x=0\implies5yz(1-3x^2)=0\implies x=\pm\dfrac1{\sqrt3}$
$5xz+2\left(-\dfrac{15xyz}2\right)y=0\implies5xz(1-3y^2)=0\implies y=\pm\dfrac1{\sqrt3}$
$5xy+2\left(-\dfrac{15xyz}2\right)z=0\implies5xy(1-3z^2)=0\implies z=\pm\dfrac1{\sqrt3}$

So ultimately, we have 9 critical points - 1 at the origin (0, 0, 0), and 8 at the various combinations of $\left(\pm\dfrac1{\sqrt3},\pm\dfrac1{\sqrt3},\pm\dfrac1{\sqrt3}\right)$ , at which points we get a value of either of $\pm\dfrac5{\sqrt3}$ , with the maximum being the positive value and the minimum being the negative one.

5 0

3 years ago

Ross has to fill up a 300 gallon tank. There are already 75

stepan [7]

Answer: He needs 9 hours to finish the task.

Step-by-step explanation:

Hi, first we have to subtract the gallons of water already in the tank (75) to the total capacity of the tank (300).

300-75 = 225

He needs to fill 225 gallons.

Since the pump Ross uses can fill 25 gallons of water per hour, the proportion is :

25 g / 1h

For 225 gallons:

225 g / x hours

Solving for x

25/1 = 225/x

x= 225/ (25/1)

x = 9 hours

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3 years ago

Classify Angle 1 & Angle 2 by choosing the correct term from the choices below.

valkas [14]

Answer:

Vertical

Step-by-step explanation:

Intersecting lines form Vertical angles.

6 0

2 years ago

Please help me find the x

xxTIMURxx [149]

Answer:

x=2

bd/ba=be/bc

6/6+4=x+1/x+1+x

6/10=x+1/2x+1

12x+6=10x+10

12X-10X=10-6

2X=4

X=4÷2=2

5 0

3 years ago

Read 2 more answers

Find the values of x and y. x= y=

mezya [45]

Answer:

7x+90+20=180

7x+110=180

7x=180-110

7x=70

7x/7 = 70/7

X=10

Y=40

7 0

3 years ago