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A circle centered at A with radius 10 is externally tangent to a circle centered at B with radius 7. A line that is externally tangent to both circles is drawn, where both circles lie on the same side of the line. This line intersects line AB at C. Find the length BC.

r₁ = 10       

r₂ =  7

10 + 7 = 17  ( hypotenuse )

10 - 7 = 3     ( short leg )

 ∠BCM = ∠ ABN            sin(∠ ABN ) = 3 / 17         ∠ ABN = 10.16424862°

BC = BM / sin(∠ ABN) = 39.6666667 units  

sin A  =   BC / BA

We  need  to  find BA.....this is  = √[8^2 + 4^2 ]  = √80  =  4√5

So

sin A  =  4 / [ 4√5]  =   1  / √5

Rationalizing the denominator we  have   sin A  =   √5 / 5

              C

                         7

D          15                    E =  38°

We  can use the Law of Cosines to solve this

CD  = √ [ CE^2  + DE^2  -  2(CE * DE)* cos (38°)  ]

CD  =  √ [ 7^2 + 15^2   - 2 (7 * 15)* cos (38°) ]

CD  = √ [ 49 + 225  - 210cos(38°) ]

CD =  √[274 - 210cos(38°) ] 

CD ≈ 10.42

is either one of those answers right? please help

The guy (hypotenuse)  and the tower and the ground form a RIGHT triangle.

Find the height of the tower    sin 32 = opp/hyp = height/guy

                                                  .5299 = height/45               height = 23.85 ft tall

Now use the 100 foot guy

sin (angle) = 23.85/100 = .2385      arcsin (.2385) = 13.8 ^o

I can't tell if sin a = √5/5 or just √5

See here :

https://web2.0calc.com/questions/can-someone-check-my-answer_1

Oh thank you so much! Thank you for helping me see the problem from different perspective and helping me solve it out, I really appreciate it!!! I've been stuck on this problem for days and I'm so happy I finally managed to get past it. Thank you!

1. Not sure if this is the fastest way.

(l+2)(h+2)(w+2)=lwh+2(lh+lw+wh)+4(l+w+h)+8

120+2(80)+4(15)+8=348

2. Hint: The base is an equilateral triangle. Try to drop the altitude.

I believe  that  EP's answer is correct   !!!!

Here's a graph  :  https://www.desmos.com/calculator/m73xbuxuph

Note  that  the  center  of  this  ellipse  is  the  midpoint   between  (-5,8) and (  3, - 4)

Thus.....the center  is    ( -1, 2)

I'm sorry I only just started using this website.  I saw other people postng multiple questions so I had no idea I wasn't allowed to do that.  Well thanks for the help...  Have a good day.

Do you have your output set to present all things enclosed in dollar signs as LaTex?

It does not work that way for most of us.

If you want your latex to present properly you must put it in the LaTex box (found in the ribbon) and you must get rid of the dollar signs.

I am not sure of this answer:

An ellipse is a graph of points where the sum of the distances from two foci is a constant....

this looks like the equation for two distances from two foci

first foci  would be 3 , -4        other is   -5, 8

distance between these points is      (3- -5)^2  +  (-4 - 8)^2 = d^2

                                                              64       + 144   = d^2                      d = sqrt 208      

First find the length of the hypotenuse using pythagorean theorem       c²  = a²+b²

^{then draw the triangle}

sin a  =  opposite / hypotenuse        where ' opposite'   is the side leg that is opposite angle A

Don't you even read your responses?

Some are getting flagged automatically. 

They are not readable, not easily anyway.

You should not be asking so many questions anyway. 

It indicates that you just want people to do your homework for you.

AND you should never ask more than one on one post.

Are you even too lazy to present them in a readable format?

Thanks for the help.  

The vertex is  (-1,8)

The  y intercept  = (0,7)

This  means  that  c   = 7

The  x coordinate  of the  vertex =    -b / [ 2a]  =  -1

This implies that    b =  2a

So  we  have  that [ using (-1,8)  ] 

8 =  a(-1)^2  + 2a(-1)  +  7

8  = a  - 2a  + 7

8  = -a + 7

1 =  -a

-1  =a

And  b = 2(-1)  = -2

So....the form is

y = -x^2  - 2x  + 7

So

a + b +  2c   =

-1 -2  + 2(7)  =

-3  + 14

11

If  the  point  (1,0) is  on the graph, then by  symmetry  with the veryex, the  point  (5,0)  is also on the  graph

Then  x  =1  and  x  = 5  are roots

So......we  have this form

y = a (x - 1) ( x - 5)

And since the point (3, 2)  is on the graph, we can solve for  "a" thusly

2 = a( 3 - 1) (3 - 5)     simplify

2 = a (2) (-2)

2= - 4a       divide  both sides  by  -4

2/- 4  =  a =   -1/2

So  we  have that

y = (-1/2)(x - 1) (x - 5)

y = (-1/2) (x^2  - 6x  + 5)

y = (-1/2)x^2  + 3x  - 5/2

(a, b, c) =   [ (-1/2) ,  3,  ( -5/2)  ]

Here's  the graph : https://www.desmos.com/calculator/vbzsmxqvnq

Let's label the angles a,b,c,d,e with a being the smallest and e being the largest. a+x = b. a+2x=c. a+3x=d.a+4x=e. A is 4x, or 44 degrees smaller than E. This means x is 11. And, if you add up all the angles, you get 540 because a pentagon is 540 total degrees. 5a+110= 540. 5a= 430. a=86. The smallest angle is 86.

Let the  measure  of  the  smallest  angle  =  A

Let the measure  of the largest  angle  be =  A  + 4D     

Where D  is the  common difference in the arithmetic sequence

And  we  know  that

(A + 4D)  - A   = 44     simplify

4D =  44   divide  both sides  by 

D  = 11

And the sum of the interior angles =  540°.....so we have that

A  + (A + 11)  + (A + 22)  + (A + 33)  + (A + 44)   = 540

5A  + 110  =  540       subtract 110 from both sides

5A  = 430     divide  both sides  by  5

A =  86°