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Mariulka [41]
3 years ago
8

Use lagrange multipliers to find the maximum volume of a rectangular box that is inscribed in a sphere of radius r

Mathematics
1 answer:
vesna_86 [32]3 years ago
7 0

For the derivative tests method, assume that the sphere is centered at the origin, and consider the circular projection of the sphere onto the xy-plane. An inscribed rectangular box is uniquely determined

1

by the xy-coordinate of its corner in the first octant, so we can compute the z coordinate of this corner by

x2+y2+z2=r2 =⇒z= r2−(x2+y2). Then the volume of a box with this coordinate for the corner is given by

V = (2x)(2y)(2z) = 8xy r2 − (x2 + y2),

and we need only maximize this on the domain x2 + y2 ≤ r2. Notice that the volume is zero on the boundary of this domain, so we need only consider critical points contained inside the domain in order to carry this optimization out.

For the method of Lagrange multipliers, we optimize V(x,y,z) = 8xyz subject to the constraint x2 + y2 + z2 = r2<span>. </span>

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Avery can run at 10 uph. The bank of a river is represented by the line 4x + 3y = 12, and Avery is at (7, 5). How much time does
Reil [10]
The minimum distance will be along a perpendicular line to the river that passes through the point (7,5)

4x+3y=12

3y=-4x+12

y=-4x/3+12/3

So a line perpendicular to the bank will be:

y=3x/4+b, and we need it to pass through (7,5) so

5=3(7)/4+b

5=21/4+b

20/4-21/4=b

-1/4=b so the perpendicular line is:

y=3x/4-1/4

So now we want to know the point where this perpendicular line meets with the river bank.  When it does y=y so we can say:

(3x-1)/4=(-4x+12)/3  cross multiply

3(3x-1)=4(-4x+12)

9x-3=-16x+48

25x=51

x=51/25

x=2.04

y=(3x-1)/4

y=(3*2.04-1)/4

y=1.28

So now that we know the point on the river that is closest to Avery we can calculate his distance from that point...

d^2=(x2-x1)^2+(y2-y1)^2

d^2=(7-2.04)^2+(5-1.28)^2

d^2=38.44

d=√38.44

d=6.2 units

Since he can run at 10 uph...

t=d/v

t=6.2/10

t=0.62 hours  (37 min 12 sec)

So it will take him 0.62 hours or 37 minutes and 12 seconds for him to reach the river.




8 0
3 years ago
PLS HELP ME, PLS THIS IS DUE FIRST CORRECT ANSWER GETS BRAINLIEST​
zepelin [54]

Answer: C. 29 3/4

Ok done. Thank to me :>

8 0
2 years ago
Read 2 more answers
Ok ong i'm struggling and need help i've been up all night- Ill give brainlyest!!
Ipatiy [6.2K]

Answer:

  1. 1
  2. - 9
  3. 13
  4. 15

Step-by-step explanation

- 7(6x-1)+x=36

  1. => 42x - 7 + x = 36
  2. => 43x - 7 = 36
  3. => 43x =  36 + 6 = 43
  4. => x = 43 / 43 = 1

- 11-2(8+3p)=7²

  1. => 11 - 16 - 6p = 7^2
  2. => 11 - 16 - 6p = 49
  3. => -5 - 6p = 49
  4. => -6p = 49 + 5 = 54
  5. => p = (54) / (-6) = - 9

-  1/4(5b+11)=19

  1. => (5/4)b +  (11/4) = 19
  2. => (5/4)b = 19 - (11/4) = 65 /4
  3. => b = (65/4) / (5/4) => (65/4) * (4/5) = 13

- 2/7(4m-18)=12

  1. => (8/7)m - (36/7) = 12
  2. => (8/7)m = 12 + (36/7) = 120/7
  3. => m = (120/7) / (8/7) = 120 /8 = 15

3 0
3 years ago
Suppose that your net income for the month is 1500 and that your total expenses are 2500 which statement is accurate?
mestny [16]

C. you have a deficit of 1000

Explanation:

We say that we have a surplus when the net income is more than the total expenses, while we say that we have a deficit when the net income is less than the total expenses.

In this case, the net income is 1500, which is less than the total expenses (2500): so, we have a deficit. In order to calculate the deficit, we can use the formula

deficit = total expenses - net income

Substituting the data of the problem, we find

deficit = 2500 - 1500 = 1000

7 0
3 years ago
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Four more than three times a number is greater than twenty-two
ololo11 [35]
4 + 3a > 22
3a > 18
a > 6
3 0
3 years ago
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