If there are 3 known functions, namely:
X₁ + 2X₂ + 3X₃ = 6
2X₁ - 2X₂ + 5X₃ = 5
4X1 - X₂ - 3X₃ = 0
Use the Jacobian determinant to see whether there is a functional freedom function for each pair. Determine the values of X1, X2 and X3 in the above equation?

Find the perimeter of the curve for one full rotation.

x=6cost−2cos3t

y=6sint−2sin3t

Please provide a detailed explanation of this problem. show the necessary formulas.

Find the centroid (¯x,¯y) of the region bounded by:
y=3x² + 7x,   y=0,   x=0,   and   x=6

Thanks.

The manufacturer of a brand of mattresses will make x hundred units available in the market when the unit price is

p = 150 + 60e^0.05x

dollars.

(a) Find the number of mattresses the manufacturer will make available in the market place if the unit price is set at $350/mattress. (Round your answer to the nearest integer.)


(b) Find the producers' surplus if the unit price is set at $350/mattress. (Round your answer to the nearest dollar.)
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1) finding the volume of solid whose upper limit is the surface f (x, y) = 4xe^y and which lower limit is the region r. where r is the triangle limited by y = 2x; y = 2; x = 0.

Evaluate or solve the following

A) dy/dx= -(2x²+y²)/(2xy+3y²)

B)dy/dx=(1+y²)/(1+x²)xy

C) (x²+1)dy/dx+2xy=4x² given that when x=3,y=4

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Find the mass and center of mass of the solid E with the given density function ρ.

E is the tetrahedron bounded by the planes

x = 0,

y = 0,

z = 0,

x + y + z = 2;

ρ(x, y, z) = 3y.

PLEASE TYPE!!

Think about where you have noticed circles in your everyday life and find at least 2 examples of circles in your everyday life. For each example, include the following in your post. Be sure to include enough details in your descriptions and explanations so someone who is not familiar with your everyday life will understand them.

1. Describe the circle's use and its measurement (radius or diameter, either an exact value or an approximation is fine).
2. If important, describe the location of the circle (analogous to our finding a circle's center).
3. Explain what you feel is important about the mathematics (measurement and/or location) about the circle.

Find the first four nonzero terms in a power series expansion about x=0 for a general solution to the given differential equation.

(x^2 +5)y"+y=0

convergent or divergent

infinity sigma n = 1 sqrt(n^5+ n^3 -7) / (n^3-n^2+n)

Let U = {(x1,x2,x3,x4) ∈F4 | 2x1 = x3, x1 + x4 = 0}.

(a) Prove that U is a subspace of F4.

(b) Find a basis for U and prove that dimU = 2.

(c) Complete the basis for U in (b) to a basis of F4.

(d) Find an explicit isomorphism T : U →F2.

(e) Let T as in part (d). Find a linear map S: F4 →F2 such that S(u) = T(u) for all u ∈ U.

Solve the recurrence relation with the given initial conditions.

b₀ = 0, b₁ = 4, b_n = 2b_{n ? 1} + 2b_{n ? 2} for n ? 2

A city council consists of eight Democrats and eight Republicans. If a committee of six people is? selected, find the probability of selecting four Democrats and two Republicans.

he function

​f(x)equals=0.030.03xplus+500500

represents the rate of flow of money in dollars per year. Assume a​ 10-year period at

88​%

compounded continuously. Find​ (A) the present​ value, and​ (B) the accumulated amount of money flow at

tequals=10.

​(A) The present value is

​$nothing