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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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8^x=2 write down the value of x
e-lub [12.9K]

Answer:

x = \frac{1}{3}

Step-by-step explanation:

Using the rule of exponents

a^m^{n} = a^{mn}

Note that 8 = 2³ and 2 = 2^{1}, then

(2^3)^{x} = 2^{1}, that is

2^{3x} = 2^{1}

Since the bases on both sides are equal, equate the exponents

3x = 1 ( divide both sides by 3 )

x = \frac{1}{3}

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3 years ago
Which of the following circles lie completely within the fourth quadrant?
Sladkaya [172]

Answer:

C. (X-9)^9 + (y+9)^2= 16

D. (X-9)^2 + (y+5)^2= 9

Step-by-step explanation:

The formula for a circle is

(X-h)^2 + (y-k)^2= r^2

where (h,k) is the center of the circle and r is the radius

The 4th quadrant is where x is positive and y is negative

Add r to the y coordinate of the center and if it is still negative, the circle is still completely in the 4th quadrant

A. (X-12)^2 + (y+0)^2= 72

The center is at 12,0 and the radius is sqrt(72) = 6sqrt(2)

This will be positive so it goes into the 1st quadrant

B. (X-2)^2 + (y+7)^2= 64

The center is at 2,-7  and the radius is 8

-7+8=1 so it goes into the 1st quadrant

C. (X-9)^9 + (y+9)^2= 16

The center is at 9,-9  and the radius is 4

-9+4 = -5  so it is completely in the 4th quadrant

D. (X-9)^2 + (y+5)^2= 9

The center is at 9,-5 and the radius is 3

-5+3 = -2 so it is completely in the 4th quadrant

4 0
3 years ago
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28

Step-by-step explanation:

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and add all the sides of the bigger triangle

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2 years ago
Find the slope of the line that contains the points named.
8090 [49]

Answer:

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Step-by-step explanation:

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3 years ago
Describe how to compare the two ratios 5:8 to 7:9. Tell each step to set up the ratios, then explain what to do to be able to co
Lera25 [3.4K]

Step-by-step explanation:

The ratio's can be expressed as fractions and you may compare those.

  • 5 : 8 = 5/8
  • 7 : 9 = 7/9

In order to compare fractions you need to bring them into equal denominator.

8 and 9 have LCM of 8*9 = 72 since they have no common factors.

<u>The equivalent fractions with common denominator are:</u>

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<u>Now we can compare the fractions:</u>

  • 45/72 < 56/72, since 45 < 56
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2 years ago
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