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Each of these circles represents a pizza cut into 10 equal slices, and the yellow slices are the ones eaten by Larry.

If Larry ate L% of the total amount of pizzas, what is the value L of ?
 

2) The arithmetic sequence begins with 1 and a common difference of 0.2422526225 as follows:

(1, 1.24225 2622, 1.48450 5245, 1.72675 7867, 1.96901 049, 2.21126 3113, 2.45351 5735, 2.69576 8357, 2.93802 098, 3.18027 3602, 3.42252 6225, 3.66477 8848, 3.90703 147, 4.14928 4093, 4.39153 6715, 4.63378 9337, 4.87604 196, 5.11829 4583, 5.36054 7205, 5.60279 9828, 5.84505 245, 6.08730 5072, 6.32955 7695, 6.57181 0317, 6.81406 294, 7.05631 5563, 7.29856 8185, 7.54082 0807, 7.78307 343, 8.02532 6052, 8.26757 8675, 8.50983 1298, 8.75208 392, 8.99433 6543, 9.23658 9165, 9.47884 1787, 9.72109 441, 9.96334 7033, 10.20559 965, 10.44785 228, 10.69010 49)

The 4th term in the 1st. Geometric Sequence appears as the 8th term in this arithmetic sequence.

\(\text{well.. the main formula they are after is}\\ P[A|B] = \dfrac{P[B|A]P[A]}{P[B]}\)

\(\text{here}\\ P[B|A] = P[3\cup 6 \cup 9]\\ P[A] = P[3\cup6\cup 9\cup 12]\\ P[B] = P[2-9]\)

\(\text{with some simple counting }\\ P[3 \cup 6 \cup 9] = \dfrac{2+5+2}{36} = \dfrac{11}{36}\\ P[3\cup 6 \cup 9 \cup 12] = \dfrac{12}{36}=\dfrac 1 3\\ P[2-9]=\dfrac{1+2+3+4+5+6+5+4}{36} = \dfrac{30}{36}=\dfrac{5}{6}\)

\(\dfrac{P[B|A]P[A]}{P[B]} = \dfrac{\dfrac{11}{36}\dfrac{1}{3}}{\dfrac{5}{6}}=\dfrac{11}{90}\)

\(\text{as north and west are perpendicular directions}\\ d = \sqrt{60^2 + 500^2} = \sqrt{253600} = 20 \sqrt{634}~km\)

no step on snek

180km/2 hours = 90km/hour

5) identify zeros and asymptotes of f(x) = (3x^2 + 2x - 8)/ (x^2 - 9)

The numerator determines the zeroes

3x^2 + 2x - 8 = 0

(3x - 4) (x + 2) = 0

Setting each factor to 0 and solving for x gives the zeroes of x = 4/3  and x = -2

We have same power polynomial / same power polynomial    situation

The horizontal asymptote comes from the ratio of the coefficients on the x^2 terms

This is

y = 3/1   =  3

The vertical asymptotes are the x values that make the denoninator = 0

Note that these are  x = 3      and x = -3

3) for the sequence 2/3 , 2/9 , 2/27 , 2/81 write the nth term formula     (Is it geometric or arithmetic? How do i know?)

Geometric....the common ratio is  1/3

Arithmetic sequences hve a common difference between terms, not a common ratio between terms

The nth term formula is     (2/3)(1/3)^(n - 1)

4.   y = 9x + 5       subtract 5 from both sides

y - 5 =  9x          divide both sides by  9

[ y - 5  ] / 9 = x            "swap" x and y

[ x - 5 ] / 9 = y          the inverse   is in red

Many people may already have looked at it.

Many people may have already been working on it.

1) what is the range of    y = 3 sqrt (x+6)

I believe this might be  y = ∛ (  x + 6)

As x assumes some large negative value........y tends toward - inf

As x assumes some large positive value......y tends toward inf

So   the range is (-inf, inf)

2) find the solution(s) to x+1 = sqrt(2x + 5) 

Square both sides

x^2 + 2x + 1 = 2x + 5       rearrange as

x^2 - 4 = 0         factor

(x + 2) (x - 2)  = 0

Setting each factor to 0 and solving for x we get that  x = -2    or  x = 2

The second solution is good

The first  is extraneous.....it makes the left side of the original equation negative....but we cannot get a negative from a positive radical

6) what is g(f(x)) if f(x) = 2x^2 - x and g(x) = x^(-1/2)

g(f(x))   just means that we are putting  "f" into "g"

So we have

(2x^2 - x)^(-1/2)  =      1 / √[ 2x^2 - x ]

And that's it...!!!

I didn't think you even saw it yet.

5) simplify (2/x) + 3/(x-1) ÷ 1/(2x-2)

Easiest to take this in "pieces"

2              3                    2          (x - 1)             2(x - 1)

_   +     ____    =        ____   *   ______  =     _______  

x           (x - 1)               x              3                    3x

So now...we have

2(x - 1)               1                    2(x - 1)          (2x- 2)            2(x - 1) * 2 (x - 1)                4(x - 1)^2

______    +     ______    =   _________  *    ______   =     _________________  =    _________

  3x                  (2x - 2)                3x                   1                         3x                                   3x

4) simplify (9x + 27)/x^2 - 6x + 8) ÷ (6x + 18)/(4x^2 - 16)

Factor tops/bottoms

9 (x + 3)                     6 (x + 3)

__________    ÷   ____________        flip the second fraction and multiply

(x -4)( x - 2)           (x + 4)(x - 4)

9(x + 3)                  (x + 4) ( x - 4)

__________    *      _____________

(x - 4) ( x - 2)           6(x + 3)

9(x + 3) * (x + 4) ( x - 4)

___________________

6  (x + 3) *  (x -4) ( x - 2)

3 (x + 4)             3x + 4

_______  =       _____ 

2 (x - 2)             2x - 4

If 2x − 2 ≤ f(x) ≤ x^2 − 2x + 2 for x ≥ 0,

find  lim x→2 f(x).

sub in x=2

\(4-2\le f(2)\le4-4+2\\ 2\le f(2)\le2\\ f(2)=2\)

We would do the same thing.....one thing though....the remaining polynomial may or may not be factorable

Fortunately....this one WAS capable of being factored....!!!!

[ We might have to use the Rational Zeroes Theorem to find the other possible roots if the remaining polynomial doesn't factor easily ]

3) solve 16^x = (1/4)^(x+4)

Note that   16 = 4^2

And 1/4 = 4^(-1)

So  we have

(4^2)^x   = (4^(-1) )( x + 4)          using  a law of exponents

4^(2x)  = (4)^ (-x - 4)      we have the same bases....so....we can solve for the exponents

2x  = -x - 4     add x to both sides

3x = - 4     divide both sides by 3

x = -4/3

So basically we take the answer of the provided factor and use that as the multiplier in the syntetic divison to get a quadratic function that we use any way to factor down into the final factors? So how would this work if there was a quadratic to the 5th power? The same thing?

radius = 25

slant height = 32

2) give x-5 is a factor of x^3 + 9x^2 - 37x - 165 what are the other factors

We can use sythetic division to dwtermine the remaining polynomial

5  [  1    9     - 37     -  165 ]

             5       70         165

     _____________________

        1   14    33           0

The remaining polynomial  is    x^2 + 14 + 33 

Factoring this we have that

(x + 11) ( x + 3)  = the remaining factors  !!!

1) find the sec θ for and angle whose teminal side contains the point (-3,4)

Sec =  r /x 

r =  sqrt [ (-3)^2 + 4^2 ]  =  sqrt (25) = 5

So....sec  θ  = 5 / - 3

The angle will be in the 2nd quadrant

To find this....use the secant inverse....

arcsec (5/-3) ≈  126.9° =   θ

Gonna eat supper, Blank.....hang on and I'll answer these in just a few minutes...

(x+ h)^2 + (y + h)^2 = (a + b)^2       expand

x^2 + 2xh + h^2 + y^2 + 2yh + h^2 = a^2 + 2ab + b^2

By the Pythagorean Theorem.....

(x^2 + h^2) + ( y^2 + h^2)  = a^2 + b^2

So....we have left that

2xh + 2yh   =  2ab

xh + yh = ab

h (x + y) = ab       multiply both sides by 1/2

(1/2) (x + y)  * h =  (1/2)ab

Since this is a right triangle    (1/2) product of legs a, b  = the area of ABC = the right side

And the left side is (1/2)base of ABC * height = the area of ABC

So...the left side = the right side

OK...here's a "made up" problem

a + bc =  5          subtract  bc from both sides

a =  5 - bc

Thanks, Guest!!