1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
snow_tiger [21]
3 years ago
11

Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in t

he plane x + 2y + 3z = 9
Mathematics
1 answer:
ratelena [41]3 years ago
8 0
The point (x,y,z) on the plane x+2y+3z=9 determines the volume of the box, since V(x,y,z)=xyz. Restricting the box to lie within the first octant is to say that x,y,z>0.

Let's do it via Lagrange multipliers. The Lagrangian is

L(x,y,z,\lambda)=xyz+\lambda(x+2y+3z-9)

with partial derivatives (set equal to 0)

L_x=yz+\lambda=0
L_y=xz+2\lambda=0
L_z=xy+3\lambda=0
L_\lambda=x+2y+3z-9=0

We have

L_y-2L_x=xz-2yz=z(x-2y)=0\implies z=0\text{ or }x=2y
L_z-3L_x=xy-3yz=y(x-3z)=0\implies y=0\text{ or }x=3z
2L_z-3L_y=2xy-3xz=x(2y-3z)=0\implies x=0\text{ or }2y=3z

We already assume x,y,z>0, so we can ignore those options, leaving us with x=x, y=\dfrac x2, and z=\dfrac x3. Substituting into the plane equation, we get

x+2\dfrac x2+3\dfrac x3=3x=9\implies x=3\implies y=\dfrac32\text{ and }z=1

So the box with largest volume has its vertex (the one opposite the vertex at the origin) in the plane at \left(3,\dfrac32,1\right), giving a volume of \dfrac92.
You might be interested in
How do you solve this problem?
babymother [125]
Let's solve your equation step-by-step.<span><span><span>16</span><span>(<span>a−4</span>)</span></span>=<span><span>13</span><span>(<span><span>2a</span>+4</span>)</span></span></span>Step 1: Simplify both sides of the equation.<span><span><span>16</span><span>(<span>a−4</span>)</span></span>=<span><span>13</span><span>(<span><span>2a</span>+4</span>)</span></span></span><span><span><span><span>(<span>16</span>)</span><span>(a)</span></span>+<span><span>(<span>16</span>)</span><span>(<span>−4</span>)</span></span></span>=<span><span><span>(<span>13</span>)</span><span>(<span>2a</span>)</span></span>+<span><span>(<span>13</span>)</span><span>(4)</span></span></span></span>(Distribute)<span><span><span><span>16</span>a</span>+<span><span>−2</span>3</span></span>=<span><span><span>23</span>a</span>+<span>43</span></span></span>Step 2: Subtract 2/3a from both sides.<span><span><span><span><span>16</span>a</span>+<span><span>−2</span>3</span></span>−<span><span>23</span>a</span></span>=<span><span><span><span>23</span>a</span>+<span>43</span></span>−<span><span>23</span>a</span></span></span><span><span><span><span><span>−1</span>2</span>a</span>+<span><span>−2</span>3</span></span>=<span>43</span></span>Step 3: Add 2/3 to both sides.<span><span><span><span><span><span>−1</span>2</span>a</span>+<span><span>−2</span>3</span></span>+<span>23</span></span>=<span><span>43</span>+<span>23</span></span></span><span><span><span><span>−1</span>2</span>a</span>=2</span>Step 4: Multiply both sides by 2/(-1).<span><span><span>(<span>2<span>−1</span></span>)</span>*<span>(<span><span><span>−1</span>2</span>a</span>)</span></span>=<span><span>(<span>2<span>−1</span></span>)</span>*<span>(2)</span></span></span><span>a=<span>−4</span></span>


7 0
3 years ago
Read 2 more answers
Both (x−3)(x+3) and (3−x)(3+x) contain a sum and a difference and have only 3 and x in each factor. If each expression is rewrit
Ierofanga [76]

The expression is rewritten in standard form and they are not the same due to the negative.

<h3>What is simplification of an expression?</h3>

Usually, simplification involves proceeding with the pending operations in the expression.

Simplification usually involves making the expression simple and easy to use later.

Both (x−3)(x+3) and (3−x)(3+x) contain a sum and a difference and have only 3 and x in each factor.

Let see the equation formed by the factor;

(x−3)(x+3)

= x^2 + 3x - 3x - 9

= x^2 - 9

(3−x)(3+x)

= 9 + 3x - 3x - x^2

= 9 - x^2

The expression is rewritten in standard form and they are not the same due to the negative.

Learn more about an expression here:

brainly.com/question/1249625

#SPJ1

5 0
1 year ago
A. Use the roster method to write the given set. You may need to consult a reference, such as the Internet or an encyclopedia. (
jarptica [38.1K]

Answer:

Following are the responses to the given question:

Step-by-step explanation:

For point A:

\{0,1,2,3,4,5,6,7,8\}

For point B:

{x | x is the positive square number less than or equal to 81}

For point C:

All given sets were equal and equal, that since elements and the element no. are identical.

7 0
3 years ago
If a hiker starts at sea level and increases her elevation at a rate of 1 foot per minute what will her elevation be in 20 minut
sattari [20]

Answer:

20 feet above sea level.

Explanation:

This can be solved by looking at the Rate of Change. It is given that for ONE minute that passes by, her elevation increases by ONE foot.

With this rate, after 20 minutes, the elevation can be shown as:

20 min × 1 foot= 20 feet.

4 0
2 years ago
Read 2 more answers
Please help this is past due
Rom4ik [11]

Answer:

See below.

Step-by-step explanation:

Alexi originally has $43 in his bank account and deposits $7 per week.

Week 1: $50

Week 2: $57

Week 3: $64

Week 4: $71

Week 5: $78

The pattern is that it increased $7 every week starting from $43.

A pattern rule would be: y = 7x + 43 where x represents the number of weeks.

7 0
2 years ago
Other questions:
  • Triangle ABC with vertices A(4, −6), B(2, −8), and C(−10, 4) is dilated by a scale factor of 2 to obtain triangle A′B′C′. Which
    13·1 answer
  • Work out the value of 5x - 6 when x= 2
    15·2 answers
  • 3. Sherri bought a package of pens that contained 15 pens. How many packages should she
    7·1 answer
  • 3-5 converts each unit.
    7·2 answers
  • Help please, 100 points
    14·2 answers
  • Subtract the following polynomials
    7·1 answer
  • Prove each of the following statements about integers by proving thecontrapositive of the statement.A. If 3nis even, thennis eve
    13·1 answer
  • Which of the following is the solution to this system of equations?
    10·1 answer
  • Gabriella walks every 3 days Mona walks every 5 days. How many days until they walk again on the same day?
    14·1 answer
  • A procedure has an 81% success rate. The procedure is tried on 12 patients what is the probability that at least 8 patients show
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!