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The smallest number that can be formed is 4 + 6 + 8 = 18, and the largest: 150 + 152 + 154 = 456.

Note how every even number up to 456 can be formed: for example, 20 = 18 + 2 = 4 + 6 + 8 + 2 = 4 + 6 + 10... 164 = 4 + 6 + 154

166 = 4 + 8 + 154... 168 = 6 + 8 + 154... 170 = 6 + 10 + 154... so on- until you reach the largest number that can be formed.

Okay, so now we know that all the even numbers 18 to 456 inclusive can be formed: how many numbers are in this list?

# of numbers {18,..., 456} = # of numbers {18 - 18,...,456 - 18 = 438} = {0/2,...,438/2} = {0,...,219} = {0 + 1,...,219 + 1} = 1 through 220 inclusive.

So, the answer is 220.

If you know your basic trig laws, you will notice that LAW OF COSINES is perfect for this problem!

What is law of cosines? In a triangle ABC, with side lengths a, b, c (each opposite to the point), the formula is that:

c^2 = a^2 + b^2 - 2ab*cosC ... where angle C is included between a and b, and c is opposite point C. Without loss of generality, you can "mix and match" a, b, c.

We know DA = 3, and since triangle DEF is equilateral, then summing DA + AF = DF = DE = 5.

Using law of cosines:

\(AE^2 = AD^2 + DE^2 - 2*AD*DE*\cos{D} = 9 + 25 - 2*3*5*\cos{D}\)

D = 60 degrees, so cosD = cos(60 degrees) = adjacent/hypotenuse = 1/2

\(AE^2 = 9+25-2*3*5*{1\over{2}}=9+25-15=19\)

Square rooting both sides, you have AE = \(\sqrt{19}\)

\(f(x)\cdot{g(x)} = (-7x^4 - 3x^2 + 2)(2x^8-3x^7+5x^6+3x^5-11x^4+3x^3-18x^2+17x-5)\)

Leading term is the term with the highest degree:\(-7x^4\cdot{2x^8}=-14x^{12}\)

Thus, the leading coefficient is -14

The (highest) degree of the function is 12

We know that the constant term must stem from the product of the constants of the original function. 2 * -5 = -10

Coefficient of x^6 would be -3x^2 * 3x^3 + 2*5x^6 + -7x^4 *-18x^2 = 7*18 - 9 + 10 = 127

Hypotenuse = 3sqrt(15)

Shorter Leg = 3 sqrt(6)

By pythagorean theorem, Longer Leg = 9

Let the point between the hypotenuse and the shorter leg be A. We know that point A must have the largest acute angle in the triangle by construction.

Using any trig function (preferably a easy-calculation one), such as sinA, we can use the inverse function of it to find the angle.

sinA = opposite/hypotenuse = 9/3sqrt(15) = 3sqrt(15)/15 = sqrt(15)/5

Then angle A = arcsin(sqrt(15)/5) = 0.886 radians = 50.8 degrees.

Let the unit circle's center be P, and the circle with radius 4, Q (the center). 

AP = 1 because AP is the radius of the unit circle. BQ = 4 with the same logic. With a translation (mapping of points) to AB, P becomes point A, and Q is moved along the line BQ by 1. Let these points be P' = A, and Q'. PAB is a 90 degree angle because the radius is always perpendicular to the tangent, and QBA is 90 degrees. Thus P'BQ' is a 90 degree triangle with P'BQ' = 90 degrees, BQ' = 3, and P'Q' = 5. By pythagorean theorem, AB = 4.

Let the midpoint of YZ = point M. Let the intersection points from M to XY and XZ be points P and Q respectively.

Let the radius of the semicircle be r, and also keep in mind that point M is also the center of the semicircle.

XMZ is a 45 - 45 - 90 because XM bisects angle X and is perpendicular to YZ, so XM = 10/sqrt(2) = 5sqrt(2)

Note how MP = MQ, which create 45 - 45 - 90 triangles: XPM and XQM. Thus, MP = MQ = r = 5sqrt(2)/sqrt(2) = 5

A semicircle is half the area of a circle: circle has area \(\pi r^2 = \pi 5^2 = 25\pi\), so the area of the semicircle = \(25\pi/2\)

Say the center of the inscribed circle is C      let the intersection of line OP that intersects minor arc AB be point P

and also let the radius of the inscribed circle be "r", a variable.

We know that PC = r, and that CO bisects angle O. Additionally, CO is the diagonal of a square with side length r, because the perpendicular from C to AO = r, and creates a 45 - 45 - 90 triangle. As a result, CO = \(r\sqrt{2}\)

Summing the line segments, \(PO = r + r\sqrt{2} = 1\) and to solve this equation:

\(r + r\sqrt{2} = r(1 + \sqrt{2}) = 1\)

\(r = {1\over{1 + \sqrt{2}}} = {1(1 - \sqrt{2})\over{(1)^2 - (\sqrt{2})^2}}= {1 - \sqrt{2}\over{-1}} = \sqrt{2} - 1\) - note, the expression after the second equation sign is deduced via difference of squares.

and if you need a decimal answer for sqrt(2) - 1 = 0.414

The following statements can be deduced:

Given: 50 members can paint a 3600 square foot wall in 20 minutes. That means that if I divide the area of the wall by 9, the time would take 9 times less

50 members can paint a 400 square foot wall in 20/9 minutes. That means if I halve the number of members, it would take double the time.

25 members can paint a 400 square foot wall in 40/9 minutes. That means if I multiply the area of the wall by 10, the time would take 10 times more.

25 members can paint a 4000 square foot wall in 400/9 minutes.