Please I need help ASAP

Use Lagrange multipliers to find the volume of the largest rectangular box in the first octant with three faces in the coordinat

dimaraw [331]

I assume the given plane has equation

x + 3y + 4z = d

for some constant d ≠ 0.

You want to maximize the function V(x, y, z) = xyz subject to the above constraint. The Lagrangian would be

L(x, y, z, λ) = xyz - λ (x + 3y + 4z - d)

Find its critical points:

∂L/∂x = yz - λ = 0

∂L/∂y = xz - 3λ = 0

∂L/∂z = xy - 4λ = 0

∂L/∂λ = x + 3y + 4z - d = 0

The first three equations tell you that λ = yz = xz/3 = xy/4. Then, for instance,

yz = xz/3 → 3yz - xz = z (3y - x) = 0 → z = 0 or 3y - x = 0

If z = 0, we get zero volume which is not useful. So 3y = x. Similarly, you would find x = 4z and 3y = 4z, so a critical point occurs when x = 3y = 4z.

In the fourth equation, we get upon substituting

x + 3y + 4z = 3x = d → x = d/3, y = d/9, z = d/12

Then the largest volume of this box would be

V(d/3, d/9, d/12) = d ³ / 324

6 0

3 years ago

What is 0.4 as a decimal

kirza4 [7]

Answer: 0.40 and 40%

7 0

3 years ago

Read 2 more answers

This morning, the temperature rose 2 degrees in 2/3 of an hour at what rate did the temperature rise per hour

insens350 [35]

Answer:

3 degrees per hour

Step-by-step explanation:

2 degrees in 2/3 of an hour

2/2 = 1

1 degree in 1/3 of an hour

1*3 = 3

3 degrees in 1 hour

5 0

3 years ago

There are 25 players in a ping-pong tournament. Each player plays every other player exactly once. How many games will be played

Aleonysh [2.5K]

This is a round robin style tournament. Given that there are 25 people, and order doesn't matter meaning that the combination AB is equivalent to BA you must use the choose formula.

Thus the answer should be 25C2 or
25! / [(25-2)! * 2!] = 300

4 0

4 years ago

What are the coordinates of the point on the directed line segment from (-8, -10)(−8,−10) to (6, -6)(6,−6) that partitions the s

frez [133]

Answer:

(2.5, -7)

Step-by-step explanation:

Given:

Two end points of a line segment are given as:

(-8, -10) and (6, -6)

The ratio of partition is 3 : 1.

Let the line segment be AB with A(-8, -10) and B(6, -6). Let point P(x,y) divide the line segment AB in the ratio 3 : 1.

In order to find the point P, we use section formula which is given as:

$(x,y)=(\frac{nx_1+mx_2}{m+n},\frac{ny_1+my_2}{m+n})$

Where, $(x_1,y_1)\ and\ (x_2,y_2)$ are end points of line segment AB divided in the ratio m : n by the point $(x, y)$ .

Now, plug in the given values and solve for the coordinates of P. This gives,

$P(x,y)=(\frac{1\times -8+3\times 6}{3+1},\frac{1\times -10+3\times -6}{3+1})\\\\P(x,y)=(\frac{-8+18}{4},\frac{-10-18}{4})\\\\P(x,y)=(\frac{10}{4},\frac{-28}{4})\\\\P(x,y)=(2.5,-7)$

Therefore, the coordinates of the point on the directed line segment from (-8, -10) to (6,−6) that partitions the segment into a ratio of 3 to 1 is (2.5, -7).

8 0

4 years ago