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f(x) = l 9 - x l    g(x)  = √x  - 1

a)  f ° g (1)   =     l  9  - (√1 - 1) l   =  l 9 - 0 l   = l 9 l  =   9

     f ° g(3)  =       l 9 - ( √3 - 1 l   =    l 10 - √3 l  ≈  8.268

     f ° g(5)  =       l 9 - ( √5 - 1 l   =   l 10  - √5  l  ≈  7.764

     f ° g(7)   =     l 9 - ( √7 - 1) l  = l 10 - √7 l  ≈  7.354

     f ° g(0)   =     l 9 - ( √0  - 1 ) l  =  l 10 l   =  10

b)  f ° f(2)  =   l  9  - l 9  - 2 l l  =  l 9 - l7l l  = l 9 - 7 l  =  2

     f ° f(5)  =   l 9  - l 9 - 5 l l  =   l 9 - l 4 l l  = l 9 - 4 l  =  5

     f ° f(-2)  =  l 9  - l 9 - - 2 l l  =  l 9 - l 11 l l = l 9 - 11l =   l - 2 l  =  2

c)  Definitely not.....consider  f ° f (10)  =    l 9 - l 9 -10l l  =  l 9 - l - 1 l l  =  l 9 -  1 l  = 8

But    l 10  l  =  10

a)  We can  apply a "3D" version of the Pythagorean Theorem here :

The longest stick that will fit is given by:

√[ 24^2  + 18^2 + 12^2  ]  = √1044  ≈ 32.31  in

b)  We needt to find the diagonal distance across the bottom of the box to first answer this....we can apply the Pythagorean Theorem  here :

This  is given by : √ [ 24^2 + 18^2  ] = √900  = 30

We can find the angle that the stick makes with the bottom of the box by using the inverse tangent

arctan  (  12 / 30)  =  angle  ≈  21.8°

This is the question.

Find all numbers r for which the system of congruences

\(\begin{align*} x &\equiv r \pmod{6}, \\ x &\equiv 9 \pmod{20}, \\ x &\equiv 4 \pmod{45} \end{align*}\)

has a solution.

I get r=1

Lets see....  (All the pronumerals I will use are integers)

x/20 = k remainder 9 so    x=20k+9

x/45 = t remainder 4 so     x=45t+4

so

20k+9 = 45t+4

45t-20k = 9-4

45t - 20k = 5

9t - 4k = 1

one obvious solution is t=1 and k=2

9( 1 ) - 4 (2) =1

9(1+4n)-4(2+9n)=1      (because I have added 36n and then taken 36n away again)

so a general solution is   t=1+4n   and  k=2+9n

x=20(2+9n)+9 = 40+180n+9 = 49+180n

or

x=45(1+4n)+4 = 45+180n+4 = 49+180n      Good they are the same.

Now we have x is equivalent to r mod 6

So

(49+180n)/6 = Q remainder r

49+180n = 6Q+r

48+180n+1=6Q+r

6(8+30n)+1=6Q+r

r = 1

SO SORRY 

To get from her home to a friend's house, Georgia must go past a school, then a post office:

home---school---post office---freind's house

Signs tell Georgia that the distance she walks from home to the school is 7/8 mile, and the distance from the school to the post office is 5/6 mile.

The whole trip takes Georgia 42 minutes. If Georgia can walk a mile in 18 minutes, then how many miles is the distance from the post office to her friend's house?

The probability of (\(4\leq x\leq 8\)) is just the area under the graph between x = 4 and x = 8.

P(\(4\leq x\leq 8\)) = (3+4)*(0.1)/2 + 4*0.05 = 0.35 + 0.2 = 0.55.

Area of that cross section = Area of triangle ABC.

Using Pythagoras theorem, BC = \(\sqrt{17^2-8^2} = 15\)

Area of ABC = 8*15/2 = 60 sq ft

Area of the cross section = 60 sq ft :)

The longest line segment that can be drawn inside an arc is the red line below:

Diameter = 12cm --> Radius = 6cm

Central angle of one arc = 360 deg / 3 = 120 deg

l^2 = 6^2 + 6^2 - 2(6)(6)cos120 deg (Law of Cosines)

I^2 = 72(1-cos(120)) = 108.

Which graph represents the system of inequalities?

\(\boxed{\text{Degree to radian: } \theta_{\text{rad}}=\theta_{\text{deg}}\cdot\dfrac{\pi}{180^{\circ}}}\)

\(\quad 510^{\circ}\\ =\dfrac{510^{\circ}\cdot\pi}{180^{\circ}}\\ =\dfrac{17\pi}{6}\text{rad}\)

\(\quad\sqrt{75x^3y^6}\\ =\sqrt{75}\cdot\sqrt{x^3}\cdot\sqrt{y^6}\\ =\sqrt{25}\cdot\sqrt3\cdot(x\cdot\sqrt{x})\cdot(y^3)\\ =5xy^3\sqrt{3x}\)

Therefore the fourth option is the answer.

(x^3  + x^2  - 9x  - 9) : (x + 1)  =  x^2 - 9  
 x^3  + x^2           
 ———————
             - 9x  - 9
             - 9x  - 9
             ————
                     0

x^2 - 9 = (x + 3) (x - 3)

x = -1

x = -3

x = 3

\(\begin{align} f(x)&=x^3+x^2-9x-9\\ f(x)&=x^2(x+1)-9(x+1)\\ f(x)&=(x^2-9)(x+1)\\ f(x)&=(x+3)(x+1)(x-3)\\ \end{align}\\ \therefore \text{-3, -1 and 3 are zeros of the polynomial.}\)

\(\begin{align}\dfrac{1}{4}\left(\dfrac{5}{12}x-\dfrac{3}{4}\right)& > x+\dfrac{1}{12}\\ \dfrac{5}{48}x-\dfrac{3}{16}& > x + \dfrac{1}{12}\\ 5x-9& >48x+4\\ 43x&<-13\\ x&<-\dfrac{13}{43} \end{align}\)

(x^4  + 3x^3  -  6x^2  - 12x  -  8) : (x - 3)  =  x^3 + 6x^2 + 12x + 24  remainder  64 

  x^4  - 3x^3                     

 —————

                6x^3  -  6x^2  - 12x  -  8

                6x^3  - 18x^2             

                ——————————

                               12x^2  - 12x  -  8

                               12x^2  - 36x     

                               ———————

                                                24x  -   8

                                                24x  - 72

                                                —————

                                                            64

f(3)=64

\(\quad(x+2)(-x^2+2x-6)\\ =x(-x^2+2x-6)+2(-x^2+2x-6)\\ =-x^3+2x^2-6x-2x^2+4x-12\\ =-x^3-2x-12\)

Therefore the answer is the third option.

3x(x+4)(x-6) = 0

x = 0 or x+4 = 0 or x-6 = 0

x = 0 or x = -4 or x = 6

I think the question is asking all roots(x) are real, not a.

Therefore, shouldn't we solve the general root first, then find the probability that all x are real?

I mean:

\(x^3+(2a+1)x^2+(4a-1)x+2 = 0\\ x = -2 , \dfrac{\sqrt{4a^2-4a-3}-2a+1}{2}, \dfrac{-\sqrt{4a^2-4a-3}-2a+1}{2}\)

So the question becomes probability that \(\sqrt{4a^2-4a-3}\) is real.

Since the equation is already factored, we just need to consider the three cases:

To solve this problem, we need to understand that the product of zero and any number is equal to zero. 

Case 1: 3x = 0 

3x = 0 

x = 0 

Case 2: x + 4 =0

x + 4 = 0 

x= -4 

Case 3: x - 6

x - 6 = 0 

x = 6 

The three solutions are 0, -4, and 6. 

I hope this helped,

Gavin

\(\sqrt{2x-5}+4=x\\ \sqrt{2x-5}+4-4=x-4\\ \sqrt{2x-5}=x-4\\ (\sqrt{2x-5})^2=(x-4)^2\\ 2x-5=x^2-8x+16\\ x^2-10x+21=0\\ (x-7)(x-3)=0\\ x_1=7, x_2 = 3 \)

We plug the solutions to verify:

\(\sqrt{2\cdot7-5}+4=7\\ \sqrt9+4=7\\ 3+4=7\\\) 

Seven works!

\(\sqrt{2\cdot3-5}+4=3\\ \sqrt1+4=3\\ 5\ne3\\\)  

Three doesn’t work. 

Therefore, the answer you seek is 7!

I hope this helped,

Gavin

In radians, π is also equal to 180 degrees.

510 degrees is also 2π + 150 degrees, 

150 degrees is also 5/6 π.

So the answer is 2 5/6π

Or 17/6 π

I hope this helped,

Gavin

Note: Density of copper is about 9 g/cm^3. The density calculated in your question is about 4,000,000,000x LESS than the true density of copper!!.

3. 

x^3+(2a+1)x^2+(4a-1)x+2=0

x = -2, ALL values of "a" are solutions.

x = - 1/2,  a = 7/4

x = 1/2,    a = - 3/4

Since ALL the solutions are "real", the probability is 100%!!.

Volume of prisim = 5 * 0.9 * 0.02  = 0.09  mi^3

Density =  Mass / Volume   =  894 / 0.09   ≈  9933.3   kg/mi^3

Population density  = Number. of  people  / Number of square miles  =

466488 / 97.9  ≈    4765 people  per square mile