The area of a particular rectangle is 12 times the area of a certain square, and the width of the rectangle is three times the length of a side of the square. The perimeter of the rectangle is 30 units greater than the perimeter of the square. Find the dimensiuons of both the rectangle and the square.

Derive the point-slope form of a line using the fact that similar triangles have proportional sides.

Assume that a 3D polygon is tranformed by a transformation matrix T. Assume the normal vector of the polygon is n. Can we use the formula Tn to compute the normal vector of the transformed polygon? If not, drive the correct formula.

Prove that an isometry takes a (poylgon) unicorn to a congruent unicorn.

Please Consider the function f : R -> R given by f(x, y) = (2 - y, 2 - x).

(a) Prove that f is an isometry.

(b) Draw the triangle with vertices A = (1, 2), B = (3, 1), C = (3, 2), and the triangle with vertices f(A), f(B), f(C).

(c) Is f a rotation, a translation, or a glide reflection? Explain your answer.

Using Euclid's Propositions:

1. Prove that the regular octagon is constructible.

2. Prove that the regular decagon is constructible.

Small triangle. Answer the following.

a. State a definition for small triangle that would eliminate a counterexample to ASA.

b. Explain why using this definition eliminates the possibility of a counterexample to ASA.

Suppose that ABCD is cyclic, so there is a circle through all four of its vertices. Prove that (?A) + (?C) = 180? and (?B) + (?D) = 180?.

Business

The annual average natural gas price (in dollars per million BTUs) can be approximated by the function
h(x) = 69.54e^?0.264x

where x = 8 corresponds to the year 2008.

(Data from: Energy Information Administration.)

(a) What was the average price per million BTUs in the year 2010?

(b) According to the model, in what year did the average price per million BTUs hit $3.00?

A polyhedron must meet three conditions in order to be regular:

All of the faces are regular polygons;

All of the faces are congruent (identical);

All of the vertex points/arrangements are congruent (identical).

Try to find three nonregular polyhedra, each of which violates a different one of these conditions. In other words, one of them should meet (1) and (2) but not (3); one should meet (1) and (3), but not (2); one should meet (2) and (3) but not (1). Describe each polyhedron using words:

number of faces and the polygons forming the faces

number of vertices and their vertex arrangements

number of edges

** Make sure to state which condition is being violated in each case

Find the dual problem for each of the following primal problems.

a): min z=6x1+8x2 st: 3x1+x2>=4 5x1+2x2>=7 x1,x2>=0

b): max z=8x1+3x2-2x3 st: x1-6x2+x3>=2 5x1+7x2-2x3=-4 x1<=0,x2<=0,x3 unrestricted

You are the owner of a company that produces and sells a new tool at hardware stores. During one month, when 'x' of those new tools were produced and sold, the following were the revenue and cost (in hundreds of dollars).  

R(x)= -0.06x² +13.12x + 65.8 C(x)=3.04x+ 7.6

5.Fill-in the blanks.Be sure to read the following carefully!

HINT: do NOT use the TABLE from your graphing calculator, but DO use the graphing capabilities.

a)The profit for that month was 79 hundred dollars when ..........................of the new tool were produced and sold.

b)The profit for that month was 98 hundred dollars when ...........................of the new tool were produced and sold.

c)The profit for that month was 316 hundred dollars when...........................of the new tool were produced and sold.

d)The profit for that month was 47,600 dollars when................................of the new tool were produced and sold.

Solve these linear equations over Z₅

2x+3y =4

3x+y=2

The answer is (1,4) as unique solution, please kindly explain the reasons and show your work, much appreciated!

A national chain of clothing goods stores recently sent shipments to the following stores. The number of items shipped to each store and their total wholesale cost is shown in the table below. Find the wholesale price of one of each item.

(A) cost of a pair of jeans is?

(B) cost of a jacket is?

(C) cost of a sweater is?

(D) cost of a shirt is?