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They keep doing this to you. Lol

Dang. I feel sorry for you.

Good Morning Synth. How are ya?

Come on EP.

Don't you think our guest could have and should have made that correction for him/herself ?

OR they could have asked if it was an error if they were not sure. 

How wonderful it would be if more of these question askers interacted with us.

Slight correction in Rom's last line

3w=75

w=25 weeks until pib-less.

What we have here is a failure to communicate!

z-5 = ?    is not a complete solveable mathematical equation......we need more information to find a distinct value for 'z'  or ' ? '       Sorry  

Max height t is also given by     -b/2a  of the quadratic equation>    -52/(-18) = t =  2.889 sec

 h(max) = -9 (2.889)^2 + 52(2.889) + 3 = 78.11  (feet?)

Y intercept is initial height of ball when t = 0    -9(0^2) +52(0) +3  = 3  (feet?)

As Alan showed, time in the air ends when h= 0       0 = -9(t^2)+52t+3     Using Quadratic Equation  t = 5.835 sec

f(x)=-2|x-4|+3 describe trasformations.

For x>=4  

This is the graph of -2x+8  but it is vertiacall lifted by 3 soit becomes the graph of  -2x+11

This is reflected in the line x=4 to get the other half of the graph.

z-5=z-5 

"P= (-9,8)       PR has the length of 20 units

A=(3,-1) 

find the nearest distance between A and PR"

Because the position of R is not specified I take this to mean "find the nearest distance between A and the locus of R".   

"The height of a baseball t seconds after it is hit up in the air is given by h=-9t2+52t+3. What is the maximum height of the ball?

I also need to know:

What is the y-intercept of the ball's path in the problem above AND what is its meaning in the problem?

How long was the ball in the air in the problem above?"

Thanks for that great answer guest, 

I will just try to show a little clearer what I think you have done.

First add 1 to both sides.

\(x+1 = 2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}\)

now

\(x+1 = 2 + \cfrac{1}{x+1}\\ x-1 = \cfrac{1}{x+1}\\ x^2-1 = 1\\ x^2=2\\ x=\pm\sqrt2 \\ \text{But everything is positive on the RHS so } x>-1\\ x=\sqrt2\)

I'd use the fact that lines tangent to the circle are perpendicular to the radius.

So NPM is a right triangle.

\(|MN|^2 = r^2 + |NP|^2\\ 25^2 - 49 = |NP|^2\\ 576 = |NP|^2 \\ |NP| = 24\)

click on the 2nd button.

i should appear top row 3rd column

\(\text{you only have 2 data points so we're talking a line}\\ p(w)= m w + p_0\\ m=\dfrac{42-57}{11-6} = -\dfrac{15}{5}=-3\\ 42 = -3(11)+p_0 \\ p_0 = 42+33 = 75\\ p(w) = -3w+75\\ 0 = -3w + 75\\ 3w = 75\\ w=25\\ \text{25 weeks until Mimstoon is pibless}\)

This computer summation for about 25 terms shows that x converges to sqrt(2):

c=1E100; listforeach(b,reverse(1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2), c=b + 1/c)

x =1.4142135623730950488016887242097

You  could also solve it algebraically in this way:

Solve for x:
1/(1 + x) + 1 = x

Write the left hand side as a single fraction.
Bring 1/(x + 1) + 1 together using the common denominator x + 1:
(x + 2)/(x + 1) = x

Multiply both sides by a polynomial to clear fractions.
Multiply both sides by x + 1:
x + 2 = x (x + 1)

Write the quadratic polynomial on the right hand side in standard form.
Expand out terms of the right hand side:
x + 2 = x^2 + x

Move everything to the left hand side.
Subtract x^2 + x from both sides:
2 - x^2 = 0

Isolate terms with x to the left hand side.
Subtract 2 from both sides:
-x^2 = -2

Multiply both sides by a constant to simplify the equation.
Multiply both sides by -1:
x^2 = 2

Eliminate the exponent on the left hand side.
Take the square root of both sides:
x = sqrt(2) 

Determine the largest possible integer  such that $942!$ is divisible by $60^n$

942!  mod  60^233 =0

 $x, y,$ and $z$ are positive integers such that $\gcd(x,y,z)=14$ and  $\text{lcm}(x,y,z)=630.$ What is the smallest possible value of $x + y + z$

x = 14, y =70, z =126

LCM[14, 70, 126] = 630

GCD[14, 70, 126] = 14

x + y + z =14 + 70 + 126 =210

This is how I would personally approach this problem.

#1: List all the integers from 1 to 100.

#2: Eliminate all even numbers and 1 and 100 from the list since we are only interested in "square-free" numbers bounded between 1 and 100.

#3: Eliminate multiples of the first perfect square I will test, 3^2, or 9. I am not testing 2^2, or 4 because I have already eliminate the even numbers from the list, so no multiples of 4 remain in the list right now. 

#4: Eliminate the multiples of the second perfect square I will test, 5^2, or 25. Again, I am not testing 4^2, or 16, because even numbers have already been filtered out. 

#5: Eliminate the multiples of 7^2, or 49. Only one number gets eliminated.

#6: Normally, I would eliminate the multiples of 9^2, but I already sifted out the multiples of nine, so all the multiples of 9^2 have already been eliminated from the list.

#7: There is no need to eliminate the multiples of 11^2, or 121, since that multiple is larger than the largest number we are testing, so we can stop the process here. The numbers that remain are "square-free."

In a list, they are 3,5,7,11,13,15,17,19,21,23,29,31,33,35,37,39,41,43,47,51,53,55,57,59,61,65,67,69,71,73,77,79,83,85,87,89,91,93,95,97.

Thank You!

This is the parent function   f(x)  =  l x l        reflected across the x axis, vertically stretched by a factor of 2,  translated to the right by 4 units and up 3 units

AB and CD bisect each other                                                 Given

Join ADBC                                                                               A line segment can join any two points

AE = BE    and CE = DE                                                          Definition of bisector

Angle CEA = Angle DEB                                                         Vertical angles

Triangle AEC congruent to Triangle BED                                SAS

AC = BD                                                                                  Congruent parts of congruent triangles

Angle EAC = Angle EBD                                                         Consequence of congruent triangles

Angles EAC and EBD are alternate interior angles                 Definition of alternate interior angles

AC  ll  BD                                                                                 A transversal (AB) cuts AC and BD forming equal

                                                                                                alternative angles EAC and EBD....by definition AC ll                                                                                                    BD

See my last 2 answers.