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Nice work, heureka  !!!!

Your friend has walked 5 MINUTES   ( or 1/12 of an hour) at 3 m/hr

1/12 hr  * 3 m/hr  = 1/4 mile    ahead of you when you start

1.00 $ / 120 ISK     * 2370 ISK   = $19.75

What is that supposed to mean???

Please explain.

2000   through   2999    is   one thousand 4 digit  integers with a '2' for the thousands digit

Hmmmm..... shifting this parabola right one and down five results in no REAL zeroes....only imaginary zeroes  ± 2i

Thank you both!

x^2 + 10x       + y^2 + 16 y        = k          put in standard form by completing the square for x and y

(x+5)²               + (y+8)2             =  k  + 25 + 64

    and   k + 25 + 64   = 36    (the radius²)         so k = -53

The equation of a circle is (x+a)^2+(y+b)^2=r^2

(x+a)^2 is equal to x^2+2ax+a^2

(y+a)^2 is equal to y^2+2by+b^2

The radius is 6, so right side of the equation is 36

So X^2+2ax+a^2+y^2+2by+y^2=36

a is 5 since 2ax=10x

b is 8 bc 2by is 16y

Substituting we get (x+5)^2+(y+8)^2=36

Expanding we get x^2+10x+25+y^2+16y+64

Subtract 36 from both sides

x^2+10x+y^2+16y-k=0

25+64-36=53

-k=53

k=-53

21 x 133 1/3 % = 21 * 1.33333... = 28

Lead:

When y = -(x+1)^2 + 1 is shifted 1 unit to the right, the vertex moves 1 un right as well. So in vertex form, the equation becomes y = -(x)^2 + 1. Then, shifting 5 units down with similar logic gets us y = -(x)^2 - 4.

Now we find the zeros of this equation. Expanding we get y = -x^2 - 4, so dividing by -1 gets us x^2 + 4 = 0. Can you go from here?

You are very welcome!

:P

Wrong. It's 65*5 - 4 = 321 correctly written words

Find the sum \(1*3 + 3*5 + 5*7 + 7*9 +\ldots\) for the first \(n\) terms.

\(\begin{array}{|rcll|} \hline \mathbf{1*3 + 3*5 + 5*7 + 7*9+ \ldots} &=& \mathbf{\sum \limits_{k=1}^{n} (2k-1)(2k+1)} \\ &=& \sum \limits_{k=1}^{n} (4k^2-1) \\ &=& \sum \limits_{k=1}^{n} 4k^2 - \sum \limits_{k=1}^{n} 1 \\ &=& 4\sum \limits_{k=1}^{n} k^2 - \sum \limits_{k=1}^{n} 1 \\ && \boxed{\sum \limits_{k=1}^{n} 1=n}\\ &=& 4\sum \limits_{k=1}^{n} k^2 - n \\ && \boxed{\sum \limits_{k=1}^{n} k^2=\dfrac{n(n+1)(2n+1)}{6} }\\\\ &=& \dfrac{4n(n+1)(2n+1)}{6} - n \\\\ &=& \dfrac{2n(n+1)(2n+1)}{3} - n \\\\ &=& \dfrac{2n(n+1)(2n+1)}{3} - \dfrac{3n}{3} \\\\ &=& \dfrac{n}{3}\Big( 2(n+1)(2n+1)-3 \Big) \\\\ &=& \dfrac{n}{3}\Big( (2n+2)(2n+1)-3 \Big) \\\\ &=& \dfrac{n}{3}\Big( 4n^2+6n+2-3 \Big) \\\\ \mathbf{1*3 + 3*5 + 5*7 + 7*9+ \ldots} &=& \mathbf{\dfrac{n}{3}\left( 4n^2+6n-1 \right)} \\ \hline \end{array}\)

Maxima and minima occur when the first derivative, (which is the gradient) is 0

So what is the first derivative of f(t)    ?      

Hint:  what is the opposite of integration?

Here is a graph that you might find useful.

What is 133 1/3 % of 21? With Solution

(21 / 100) * 133.3333333334 = 28 

You are going to get a face full of radioactive GingerAle!

That will be a good read too!

Ginger does write great stories and memoirs. You’d like them all. Most of them I’ve read to my youngest granddaughter, Melody Marie, who is about your age.  Ginger sent me several of her writings, including a few excerpts from the memoir she mentions to CPhill, about a chimp named Harvey. This chimp and his twin brother, Harold, jump into Ginger’s arms when she sings to them.

Part a is easy.

Let

  \(z=e^{i\theta}=cos\theta + isin \theta\\ \bar{z}=cos\theta - isin \theta\\ \\~\\ \frac{1}{z}=\frac{1}{cos\theta + isin \theta}\\ \frac{1}{z}=\frac{1}{cos\theta + isin \theta}\times \frac{cos\theta - isin \theta}{cos\theta - isin \theta}\\ \frac{1}{z}=\frac{cos\theta - isin \theta}{cos^2\theta + sin^2 \theta}\\ \frac{1}{z}=\frac{cos\theta - isin \theta}{1}\\ \frac{1}{z}=cos\theta - isin \theta\\ \frac{1}{z}=\bar z\)

Same logic for w 

------------------------

999 divided by 37 = 27

plus it could be 000

makes a total of 28  just as guest already determined.     

deleted

(37000, 37037, 37074, 37111, 37148, 37185, 37222, 37259, 37296, 37333, 37370, 37407, 37444, 37481, 37518, 37555, 37592, 37629, 37666, 37703, 37740, 37777, 37814, 37851, 37888, 37925, 37962, 37999)==28 such numbers.

Angle DEF = 100 degrees

c (1.5)  -  3   =   -5.25      add 3 to  both sides

1.5c  =  -2.25     divide  both sides by 1.5

c =   -1.5 

y = x^2   - 4x +  21

x coordinate of the  vertex      4 / (2)  =  2

y coordinate of the  vertex =   2^2  - 4(2)  + 21 =  17

The verxtex   is   ( 2,17).....above the x axis...since this parabola turns upward, there aren't any x intercepts...

However...if you mean   y   =  -x^2  -4x + 21

The  x coordinate of the  vertex is   4/ (2 * -1)  =   -2

And the y coordinate  of the vertex is  - (-2)^2  - 4(-2)  + 21  =  -4 + 8 + 21 =  25....this is the  height of the triangle

We can find the x intercepts as

-x^2 -4x +  21  =  0     multiply through by  -1

x^2  + 4x  - 21  =  0       tactor

(x   + 7)  ( x  - 3)  = 0

Set  each factor to 0 and solv for  x  and we  get  that x=   -7  and  x =  3

The  base of the  triangle  = ( 3 - - 7)   =10

So....the area is  (1/2) (base)(height)  = (1/2) (10)(25)  =   125