geno3141

Using the quadratic equation to find the solutions to  7x² -x - 3  =  0, we can get these answers:

a  =  [ 1 + sqrt(85) ] / 14       and     b  =  [ 1 - sqrt(85) ] / 14 

Both  1 + a + a² + a³ + ...     and    1 + b + b² + b³​ + ...     

are  infinite geometric series with common ratios in the range  -1 < r < 1,

so we can use the formula:  Sum  =  a₁ / ( 1 - r)

In both cases,  a₁ = 1, so:

the sum of  1 + a + a² + a³ + ...  is  1 / [ 1 - (1 + sqrt(85))/14 ]  

     =  14 / [ 14 - (1 + sqrt(85) ]     =     14 / [ 13 - sqrt(85) ]

and the sum of 1 + b + b² + b³​ + ...   is  14 / [ 13 + sqrt(85) ]

Multiplying these two values together, we get  196 / (169 - 85)   196 / 84  =  7/3

#2:  Let M be the midpoint of side AC. Draw BM.

Since triangle ABC is an isosceles triangle BM is the angle bisector of angle(B) and also the altitude from B to side AC.

This makes angle(AMB) a right angle.

Since AC = 16, AM = 8.

Using the Pythagorean Theorem on triangle(AMB), we can find that BM = 15.

In right triangle(AMB), cos(A)  =  8/17   --->   angle(A)  =  cos^-1(8/17)  =  61.9275^o.

Draw the bisector of angle(A). Call the point where this angle bisector intersects BM point 'X'.

Angle(MAX) = ½·angle(A)  =  ½·61.9275^o  =  30.96^o.

To find XM of triangle(MAX)  --->   tan(30.96^o)  =  XM/8   ---XM  =  8·tan(30.96^o)  =  4.8.

The radius of the incircle is  4.8.

I'll try the first one.

Since the triangle has side lengths of 10, 24, and 26, this triangle is a right triangle (10² + 24² = 26²).

The center of the circumcenter of a right triangle is the midpoint of the hypotenuse.

Since the hypotenuse has a length of 26, the distance from this point to any of the vertices is 13.

Therefore, the area of the circumcircle is:  pi·13²  =  169·pi.

The center of the incircle is the point of intersection of the angle bisectors.

Call the vertex between the 10 and 24 sides = C (it is a right angle).

Call the vertex between the 24 and 26 sides = A.

Call the center of the incircle X.

Draw the perpendicular from X to side AC; call this point Y; this is a radius of the incircle;

call its value 'h'.

Angle C = 90^o; therefore, angle(XCY) = 45^o.

I will need to find the size of angle(XAY).

I will use the sin(A) in triangle ABC) and find one-half of that value.

sin(A) = 10/26   --->   A  =  sin^-1(10/26)  =  22.61986^o   --->   angle(XAY)  =  11.31^o.

Now, to find the value of h:

Call the distance from C to Y 'x'; therefore, the distance from A to Y is '24 - x'.

In triangle(XCY), tan(45^o)  =  h/x   --->   h  =  x·tan(45^o).

In triangle (XAY), tan(11.31^o)  =  h/(24 - x)   --->  h  =  (24 - x)·tan(11.31^o). 

Setting these two equation equal to each other:  x·tan(45^o)  =  (24 - x)·tan(11.31^o)

--->   x·tan(45^o)  = 24·tan(11.31^o) - x·tan(11.31^o)

--->   x·tan(45^o) + x·tan(11.31^o)  =  24·tan(11.31^o)

--->   x·( tan(45^o) + tan(11.31^o) )  =  24·tan(11.31^o)

--->   x  =  24·tan(11.31^o) / [ tan(45^o) + tan(11.31^o) ]

--->   x  =  4

Therefore, the area of the incircle is:  pi·4²  =  16·pi.

169·pi  -  16·pi  =  153·pi 

Howard bought a total of  4.5 + 1  =  5.5  pounds of jellybeans.

At  $0.80  per pound, it cost him  $0.80 x 5.5  =  $4.40.

To find how high up the tower the guy wire is attached, use sine.

    sin(36^o)  =  x / 60

                x  =  60 · sin(36^o)

                x  =  35.27 ft

To find the new angle, use inverse sine.

    sin(x^o)  =  32.57 / 110

           x^o  =  sin^-1( 32.57/110)

           x^o  =  17.2^o

5 + 5*x  =  57

 5 + 5x  =  57

Subtract  5 from both sides:

        5x  =  52

Divide both sides by 5

          x  =  52/5  =  10.4

Ratio of children to adults  =  2:3

Set up the equation:  2:3  =  80:x

                                  2/3  =  80/x

Cross-multiply:          2·x  =  80·3

                                   2x  =  240

Divide by 2:                  x  =  120

Let  x  represent the county's population in 1990.

In the year 2000, the county's population was 15,922,543.

This represents an increase of  25.325.3%.

To find the original population: 

  add the original population to the increase in population to get the final population

  -  the original population  =  x

  -  the increase in population is  25.325.3% · x   =  0.253253 · x

  -  the final population is 15,922,543

                                                  x  +  0.253253·x  =  15,922.543

                                                          1.253253·x  =  15,922.543

                                                                          x  =  15,922.543 / 1.253253

                                                                          x  =  12,704,971

Distance Formula:  Distance  =  sqrt[ (x₂ - x₁)² + (y₂ - y₁)² ]

Let  (x₁, y₁)  =  (a, 21)     and   (x₂, y₂)  =  (4,7)

Distance  =  sqrt[ (x₂ - x₁)² + (y₂ - y₁)² ]     

                    --->     10·sqrt(2)  =  sqrt[ (4 - a)² + (7 - 21)² ] 

                    --->     10·sqrt(2)  =  sqrt[ (4 - a)² + (-14)² ] 

                    --->     10·sqrt(2)  =  sqrt[ (4 - a)² + 196 ] 

square both sides:

                    --->     100·2  =  (4 - a)² + 196 

                    --->        200  =  (4 - a)² + 196 

                    --->            4  =  (4 - a)² 

take the square root of both sides:

                    --->     Either  4 - a  =  2     or     4 - a  =  -2

                    --->     Either       a  =  2     or          a  =  6

Summing the two answers:  2 + 6  =  8                     

It is impossible to draw 12 yellow balls, but you could draw 11 red, 11 green, and 9 yellow without getting 12 of the same color; however, the next ball is guaranteed to give you 12 of the same color.

11 + 11 + 9 + 1  =  32