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Ber [7]
3 years ago
12

Which point of reference is the curve y3 + y = x3 + x symmetric about?

Mathematics
1 answer:
katrin2010 [14]3 years ago
8 0

Answer:

C: Origin

Step-by-step explanation:

Zero equals zero

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The base of an aquarium with given volume V is made of slate and the sides are made of glass. If the slate costs seven times as
Olin [163]

Answer:

x = ∛(2V/7)

y = ∛(2V/7)

z = 3.5 [∛(2V/7)]

{x,y,z} = { ∛(2V/7), ∛(2V/7), 3.5[∛(2V/7)] }

Step-by-step explanation:

The aquarium is a cuboid open at the top.

Let the dimensions of the base of the aquarium be x and y.

The height of the aquarium is then z.

The volume of the aquarium is then

V = xyz

Area of the base of the aquarium = xy

Area of the other faces = 2xz + 2yz

The problem is to now minimize the value of the cost function.

The cost of the area of the base per area is seven times the cost of any other face per area.

With the right assumption that the cost of the other faces per area is 1 currency units, then, the cost of the base of the aquarium per area would then be 7 currency units.

Cost of the base of the aquarium = 7xy

cost of the other faces = 2xz + 2yz

Total cost function = 7xy + 2xz + 2yz

C(x,y,z) = 7xy + 2xz + 2yz

We're to minimize this function subject to the constraint that

xyz = V

The constraint can be rewritten as

xyz - V = 0

Using Lagrange multiplier, we then write the equation in Lagrange form

Lagrange function = Function - λ(constraint)

where λ = Lagrange factor, which can be a function of x, y and z

L(x,y,z) = 7xy + 2xz + 2yz - λ(xyz - V)

We then take the partial derivatives of the Lagrange function with respect to x, y, z and λ. Because these are turning points and at the turning point, each of the partial derivatives is equal to 0.

(∂L/∂x) = 7y + 2z - λyz = 0

λ = (7y + 2z)/yz = (7/z) + (2/y) (eqn 1)

(∂L/∂y) = 7x + 2z - λxz = 0

λ = (7x + 2z)/xz = (7/z) + (2/x) (eqn 2)

(∂L/∂z) = 2x + 2y - λxy = 0

λ = (2x + 2y)/xy = (2/y) + (2/x) (eqn 3)

(∂L/∂λ) = xyz - V = 0

We can then equate the values of λ from the first 3 partial derivatives and solve for the values of x, y and z

(eqn 1) = (eqn 2)

(7/z) + (2/y) = (7/z) + (2/x)

(2/y) = (2/x)

y = x

Also,

(eqn 1) = (eqn 3)

(7/z) + (2/x) = (2/y) + (2/x)

(7/z) = (2/y)

z = (7y/2)

Hence, at the point where the box has minimal area,

y = x,

z = (7y/2) = (7x/2)

We can then substitute those into the constraint equation for y and z

xyz = V

x(x)(7x/2) = V

(7x³/2) = V

x³ = (2V/7)

x = ∛(2V/7)

y = x = ∛(2V/7)

z = (7x/2) = 3.5 [∛(2V/7)]

The values of x, y and z in terms of the volume that minimizes the cost function are

{x,y,z} = {∛(2V/7), ∛(2V/7), 3.5[∛(2V/7)]}

Hope this Helps!!!

7 0
3 years ago
234/341 in simplest form?
Natali5045456 [20]

Answer:

0.686 (approximately)

Step-by-step explanation:

8 0
3 years ago
What is the complete factorization of r2 - 6r - 9s2 + 9? A) (r - 3 + 3s)(r - 3 - 3s) B) (r + 3 + 3s)(r - 3 - 3s) C) (r - 3 + 3s)
nikitadnepr [17]
r^2-6r-9s^2+9=r^2-6r+9-9s^2=r^2-2\cdot r\cdot3+3^2-9s^2\\\\\text{use}\ (a-b)^2=a^2-2ab+b^2\\\\=(r-3)^2-9s^2=(r-3)^2-(3s)^2\\\\\text{use}\ a^2-b^2=(a-b)(a+b)\\\\=(r-3-3s)(r-3+3s)\to A
6 0
3 years ago
Using ƒ(x) = x + 6 and the domain { -1, 0, 1 }, find the range.
mars1129 [50]
The answer to the question

6 0
3 years ago
An independent-measures research study uses two samples, each with n = 12 participants.If the data produce a t statistic of t =
lana [24]

Answer:

B- It cannot be answered without additional information.

Step-by-step explanation:

From the question, we have α= 0.01 and α= 0.05.

Thus;

We can decide to look at the critical values and check if test statistic is less or more than the tabulated value for α= 0.01 and 0.05.

However, that will not be sufficient as neither the level of significance nor confidence level is clearly mentioned in the question and thus, we can not answer it without additional information.

7 0
3 years ago
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