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Thanks, EP.....Here's another way to see this :

If

1  = ( x + 12) / (x - 4)....then the numerator and denominator would have to be the same thing because   a / a  = 1    [ and a ≠ 0 ]

Thank you CPhill. Very kind and too modest of you!! I can see myself attempting to solve it using Algebra, but from your solution I don't think I could have factored "10a + b - 25" into "10(a - 2) + (b- 5)" !!!. Believe or not, I plugged it into Wolfram/Alpha and it CANNOT factor it!!!??.

I AM in !!!      Start churning out the third graders with the new nomenclature!    

Lateral edge length = e =4 = sqrt(h^2 + 1/2 a^2 )       h=height  a = side length (8----- since it is a square  32/4 = 8)

Hmmmmmm....

I would ask if you have your numbers entered correctly in your question !    Lateral edge of 4????   maybe a different value?

See the following:

http://www.calculatorsoup.com/calculators/geometry-solids/pyramid.php

Well I guess that is why I could never find it.  You would think that this symbol would have a name.  It isn't obscure and is frequently used.  I actually teach third grade and I've always called it the "rooftop".  I wonder what other teachers call it.  We want students to use the proper mathematical terms, which I'm not totally against, that you would think something they learn to use in the third grade would have a name.  I really am amazed by this.  

Thanks for your reply...maybe you and I can make our millions by coming up with a name.  I'll let you do all the legwork and I'll take the credit...does 60/40 sound fair when we receive our first million?  LOL!!!!

LOL!!!!!

Sometimes......

Logic >  Algebra  ........

(1)

x^2 + y^2  = 5    (1)

y  = 2x + k   (2)

The slope of the line is 2......usiing implicit differentiation...the slope of any tangent line to the circle is given by    ....    -x / y

So  ....equating  slopes

-x / y =   = 2     →   x  = -2y         (3)    ...... sub this into (1)

(-2y)^2 + y^2  = 5

4y^2 + y^2  = 5

5y^2  = 5     →   y  = ± 1

When y = ± 1 ....using (3),  x = - 2  or x = 2

So....we have the points      (-2, 1)  and (2, - 1)

And using (2) and solving for k........  k  =  5      or k = - 5

Here's a graph :   https://www.desmos.com/calculator/qkhdev0shy

(2)

And it's obvious that the line will intersect the circle at two points whenever 

-5 < k < 5

(3)

And  the line will not intersect the circle whenever 

k < - 5   or   k > 5

Bravo, CPhill!!.

I don't think I can solve this one using Algebra! So, here goes some "Logic!"

First, my understanding is that by "first digit", I take it to mean the "left" digit of his age, i.e., 7 in 75.

If my brother is in his 30s, then I would have to be 12, since the sum =3

If my brother is in his 40s, then I would have to be 22, since the sum =4.........and so on up to his 90s, in which case I would to be 72. In each case, the last digit of my age is 2. Therefore, the last digit of his age must be a 7.

And now it should be easy to see that the only instance where the digits of his age, when multiplied together, equal my age is when he is 67 and I am 42!.

That is my attempt!. Let us see if somebody can solve it using Algebra!.

Let your broher's age  = 10a + b    where a and b are the digits of his age 

And your age is      10a + b - 25  =  10(a - 2) + (b- 5)   where (a -2) and (b- 5)  are the digits of your age

And....the digits of his age multiplied togeher = your age

ab = 10a + b - 25     (1)

And.....the sum of he digits of your age  = the first digit of his age

(a - 2) + (b - 5)  = a  →  a + b - 7  = a  → b - 7 = 0 →   b = 7

So ......using (1) we have

a7 = 10a + 7 - 25    →

-3a  = - 18    →

a  = 6

Your brother  is 67 and you are 67 - 25  =  42

72d^2 / 20d =3.6d - width of the rectangle

Expand the following:
(x + 2) (2 x + 5) = 11 (x + 2)

(x + 2) (2 x + 5) = (x) (2 x) + (x) (5) + (2) (2 x) + (2) (5) = 2 x^2 + 5 x + 4 x + 10 = 2 x^2 + 9 x + 10:
2 x^2 + 9 x + 10 = 11 (x + 2)

11 (x + 2) = 11 x + 11 2:
2 x^2 + 9 x + 10 = 11 x + 11 2

11×2 = 22:
2 x^2 + 9 x + 10 = 11 x + 22

Subtract 11 x + 22 from both sides of 2 x^2 + 9 x + 10 = 11 x + 22:
2 x^2 + 9 x - 11 x + 22 + 10 = (11 x + 22) - 11 x + 22

(11 x + 22) - (11 x + 22) = 0:
2 x^2 + 9 x - (11 x + 22) + 10 = 0

-(11 x + 22) = -22 - 11 x:
2 x^2 + 9 x + -22 - 11 x + 10 = 0

Grouping like terms, 2 x^2 + 9 x - 11 x - 22 + 10 = 2 x^2 + (9 x - 11 x) + (10 - 22):
2 x^2 + (9 x - 11 x) + (10 - 22) = 0

9 x - 11 x = -2 x:
2 x^2 + -2 x + (10 - 22) = 0

10 - 22 = -12:
Answer: |2 x^2 - 2 x + -12 = 0

For a function f(x), it's reciprocal is defined as 1 / f(x)

So  if

f(x)  = 1/ (√(x^2  –  1)

The reciprocal function is :

1 / f(x)  =  1 / [ 1/ (√(x^2  –  1) ] =   √(x^2  –  1)

It's settled......30 machines......  a bottle for all the answerers......

If 5 machines can make 1,000 bottles in 4 days......they can make 1/4 of this =  250 bottles in one day

So....to make 1500 bottles in one day, we need :

1500/250  * 5  =

6 * 5  =

30  machines

Again with the dimensional analysis:

1000 B / (5 M x 4 D) =  50 Bottles/(Machine-Day)        50 Bottles per Machine-Day

1500 B/ (1 D x 50 B/M-D)  =  30 M     30 Machines required

5 machines can make =1,000 / 4 =250 bottles in 1 day

250/5 =50 bottles made by 1 machine in 1 day. Therefore,

1,500 / 50 = 30 machines to make 1,500 bottles in 1 day.

Only this one can be factored:

(8a^2)+16b^2 =8(a^2 + 2b^2)

Let the oldest =a
Middle child =a-1
Youngest child =a-2
a + a -1 + a - 2=23+a
2a =26
a=13 - The oldest child
Next =12- The middle child
next =11- The youngest child
23 + 13 =36 - The mother's age

google it

20th term =F + (N - 1) x D

38 = 10 + (20 -1 ) x D

38 = 10 + 19D

38 - 10 =19D

28 = 19D

D =28/19 =1 9/19 - The common difference

Sum =[F + L] /2 x N

Sum =[10 + 38] / 2 x 20

Sum =480, OR

Sum =N/2 [2F + (N-1) x 1 9/19]

Sum =20/2[20 + (19 x 1 9/19)

Sum = 480

simplify

secx-sinxtanx

50 /12 x $18 =$75 cost of 50 dozen buttons.

Circumference =140 x Pi(3.141592) =439.82288 feet - Length of the circular track.

5,280 / 439.82288 =~12 Full laps to run 1 mile.

f(x) = 1 = (x+12)/(x-4)     Multiply both sides by x-4

(1) (x-4) = x + 12             Add 4 to both sides

x = x + 16                        Subtract xfrom both sides

0 = 16       ERROR......  NOT TRUE.....This has no solution !