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This link should work ... 

AO = OB = AB =>  a

Area = 1/6 [ a² pi - a² (3√3) / 2] 

Since  AB = 6  and  BC = 10,  AC  = 8    (Pythagorean Theorem)

cos(A)  =  8 / 10.

Since  tan  =  opposite / adjacent  --  If the tan(Z) = 3 = 3/1   

--->   opposite = 3  (or a multiple thereof)     and     adjacent = 1  (or the same multiple thereof)

This means that the hypotenuse is:  h²  =  3² + 1²  =  10

--->   hypotenuse  =  sqrt(10)  (or the same multiple thereof)

cos(Z)  =  adjacent / hypotenuse  =  1 / sqrt(10) 

61)  C  =  2·pi·r  =  x

        A  =  pi·r²  =  y

Since  x  =  2y   --->   2·pi·r  =  2·( pi·r² )

                                  2·pi·r  =  2·pi·r² 

Simplifying:                       r  =  r²

                                         0  =  r² - r

                                         0  =  r(r - 1)

Solving:          either  r  =  0     or     r  =  1 

The formula for the first n terms of a geometric series is:  Sum  =  a · (1 - rⁿ) / (1 - r)

where  a  is the first term and  r  is the common ratio.

We have:  10  =  a · (1 - r¹⁰) / (1 - r)     and     70  =  a · (1 - r³⁰) / (1 - r) 

combining:  [ a · (1 - r³⁰) / (1 - r) ] / [ a · (1 - r¹⁰) / (1 - r) ]  =  70 / 10

reducing:     (1 - r³⁰) / (1 - r¹⁰)  =  7

rewriting:     ( r³⁰ - 1 ) / ( r¹⁰ - 1 )  =  7

If we divide  ( r³⁰ - 1 )   by  ( r¹⁰ - 1 )  we get  r²⁰ + r¹⁰ + 1

so:    r²⁰ + r¹⁰ + 1  =  7

and:  r²⁰ + r¹⁰ - 6  =  0

Let  x = r¹⁰   --->   x² = r²⁰

so:  r²⁰ + r¹⁰ - 6  =  0     --->     x² + x - 6  =  0     --->     (x + 3)(x - 2)  =  0

x can't be negative, so  x = 2     --->     r¹⁰  =  2     --->     r  =  2^1/10     --->     r  =  1.071773463...

Since  a · (1 - r¹⁰) / (1 - r)  =  10     --->     a · (1 - 1.071773463¹⁰) / (1 - 1.071773463)  =  10

Solving:  a  =  0.7177346254   [ or  a  =  -10(1 - 2^1/10) ]

To find the sum of the first 40 numers:  Sum  =  a · (1 - r⁴⁰) / (1 - r)  =  150

Angle o is 60 degrees (equilateral triangle)

60 / 360    x    pi  x 8^2 -   8^2 sqrt3/4                 red is area of equilateral triangle  

We have (4+1)!/4!=5 so we have 5 ways to distribute 4 identical balls among 4 identical boxes

The circle has area 64pi and that means angle AOB is 60 degrees since it's equalateral so we have 32pi/3 for the sector and for the triangle is 8*4sqrt3=32sqrt3 we then divide by 2 to get 16sqrt3 we subtract to get 32pi/3-16sqrt3 so we have \(\boxed{(B)\dfrac{32\pi}3-16\sqrt3}\)

Thanks!

Good question...

thanks EP :)

Yikes....   the second digit can only go as high as 5 ....

Here is my answer:

3 digits,

chose the first digit 9 choices      1,2,3,4,5,6,7,8,9

Chose the second 6 choices    0,1,2,3,4,5

chose the third, it has to be the same as the first so 1 choice

9*6*1=54 ways

Thanks for the correction EP.

4 digits

you can do this one yourself.

thank you!!!!!!!!!

of course they are!

(2x + 3x) - 1 = 0

5x -1 = 0

the 12 - 16 triangle has a hypotenuse of 20  (via pythag theorem)

20 is to 16      as     x  is to 20

20/16 = x/20

x = 25

In a right triangle.....one angle is 90^o      the other two add to 90^o

   in ratio of 3 to 7     one is   3 /(3+7) = 3/10^ths   of 90      3/10 x 90 = 27^o

                          and the other is   7/(3+7) = 7/10^ths   of 90 =               63^o

2 the angles add to 90

     x = one     the other is    2x -30

x    +    2x-30   = 90

 x =40          one angle is 40^o    the other   is    2x-30   =   50^o

See answer #2 here:  https://web2.0calc.com/questions/the-powers-of-coffee-in-mathematicians#r2

angle(x)  =  angle(y) + angle(z)

197 - 5n  =  ( 6n + 22 ) + ( n + 7 )

197 - 5n  =  7n + 29

       168  =  12n

           n  =  14

you get to finish ........

12 liters final

x = 10%

12-x = 100% solution

.1 x   +   1 (12-x)   ==  .3 (12)

.1x + 12 - x = 3.6

.9x =8.4

x = 9 1/3   liters of 10%

This is just a tedious exercise:

-4 + 9i     +   6 - 6i  ==  11-12i   +  (-9+15i)  

2 + 3i                     ==  2 + 3i

If  p  and  q  are inversely proportional, then there is a constant  k  such that:  p  =  k / q

If  p = 25  then  q = 6     --->     25  =  k / 6     --->     25 · 6  =  k     --->     k  =  150

For this problem, the formula is:  p  =  150 / q

Snce  q = 15     --->     p  =  150 / 15  =  10