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In order of the clues given, the linear system is:   A + B + C = 180 A = 3B A = C+16   Solve the second equation for B and the third for C both in terms of A.   B = A / 3 C = A - 16   Plug these in to the first equation...

limt→0 [1/t - 1/(t2+t)]   = limt→0 [(t+1)/(t2+t) - 1/(t2+t)]   = limt→0[t/(t2+t)]   =limt→0 1/(t+1)   = 1/(0+1)   = 1

System: (1): x+y = -1 (2): x+2y = 0 Solution: (3)=(2)-(1): (x+2y) - (x+y) = 0 - (-1)                     x + 2y - x...

System:   (1): x+3y=7 (2): 6x+21y=51   Solution:   (3)=(2)-6(1): (6x+21y) - 6(x+3y) = 51 - 6·7                        ...

A rectangle with this property is called the Golden Rectangle and the number x is called the Golden ratio, usually denoted by φ.   To solve for x, notice that an x-1 by 1 rectangle that remains is similar to a 1 by x rectangle. In similar the ratio of corresponding sides are the same...

Let W be the width. Then the length is L=3W   So The equation is then 3W·W = 50 m2 or 3W2 = 50 m2.

Let the number have the decimal representation AB.   It's value is 10A+B   The clue gives   10A + B = A + B + 27   9A = 27   A = 3   10·3 + B = 3 + B + 27   30 + B = B...

The x-intercept is where y = 0 so you get   x2 - 8x = 0   x(x-8) = 0   x = 0 or x = 8   They are at the points (0,0) and (8,0).   The y-intercept is where x = 0 so you get   y = 02 - 2·0 = 0   It is at the...

Recall that the number of ways to choose k objects from a set of n objects is the binomial coefficient   C(n,k) = n! / [k! (n-k)!]   There are n = 13 hearts in the deck and you are choosing k = 5 cards so the number of hands are   C(13,5) = 13! / (5! 8!) =...

I will set up the integrals for each.   Problem 1: x-axis and f(x)=2x-x2   Take g(x) = 0, whose graph is the x-axis.   Now let's solve the equation f(x) = g(x)   2x-x2 = 0 x(2-x) = 0   x = 0 or x = 2   So the...

The sum simplifies to ∑ zn for n=1,...,∞.   Use the ratio test   an = zn so |an+1/an| = |zn+1/zn| = |z|   Recall that if limn→∞|an+1/an| < 1, the series converges. If it's >1 the series diverges. Inconclusive if it's =1   Thus...

First of all, plugging in x=0 gives you that f(y)=f(0)+f(y) or f(0)=0.   Secondly, say that f(1) = s. Using induction it follows that f(n)=ns.   Say also that f(a) = t for some rational a ∉ {-1,0,1}. Then f(na) = nt by induction.   It also follows that...

Let S be the length of a side of the equilateral triangle. Drop an altitude from a vertex to split it into two 30-60-90 right triangles. Each has hypotenuse of length S so the short leg has length S/2 and the long leg (the altitude drawn) is (S√3)/2 So the equilateral triangle has area BH/2...

Since marbles are not replaced after being drawn, once a red one is drawn, it can't be drawn again. So the other of the two marbles drawn is one of the other three colors. There are three possible pairs of colors (Red,Blue), (Red, White), and (Red, Yellow).   If order in which the two...

Let's represent Bob's current account balance X. The clue says that if he added \$150, making the balance or X+150, that balance will be 5X. This gives us an equation to solve.   5X = X + 150   4X = 150   X = 150/4 = 37.5   He currently has...

You don't need the perpendicular distance to L.   Tables of moments of inertia usually mention the circular disk plate about the axis of rotation, which is mr2/2 Placing it on the x-y plane, this is Iz and by the perpendicular axis theorem for flat plates, Ix+Iy = Iz so Ix =...

Let n be the number of nights spent.   Since the first two nights are each \$75, if n≤2, the \$75n   Since each additional night after that, is \$50, if n>2, such a client pays \$150 for the first 2 nights and \$50 for each of the other n-2 nights for a total of \$[150+50(n-2)]...

Not that (f-g)(x) = f(x) - g(x)   f(4) = -4·44 = -1024   g(4) = 4+2 = 6   (f-g)(4) = f(4) - g(4) = -1024 - 6 = -1030

If a sequence {an} is generated by a polynomial of degree n, then the differences of consecutive terms {bn = an+1 - an} (called the first differences) form a sequence generated by a polynomial of degree n-1. The n-th differences sequence is constant   -1   2   ...

Let C be the number of chairs per row and R be the number of rows. The first clue gives CR = 54. In it plug in the equation of the second clue which is C = 2R - 3.   (2R - 3)R = 54   2R2 - 3R - 54 = 0   Factor by looking for two numbers with sum -3...