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The distances from a point on the circumcircle of an equilateral triangle to the two nearest vertices are 3 and 6.     

Find the distance from the point to the furthest vertex.     

Draw it and the answer is obvious.  There's a third of the circle's circumference from each vertex to the next.     

From the point to the two nearest vertices is 3 to one and 6 to the other.  So a third of the circumference is 9.     

You start from the point and go the direction that's 6 to a vertex, and then it's 9 to the next vertex.  6 + 9 = 15     

_.     

If c > a > 0 and a - b + c = 0, find the larger root of ax^2 + bx + c = 0.     

The first one I thought of is          x² + 3x + 2 = 0     

                                                   (x + 1)(x + 2) = 0     

                                                                      x = – 1     

                                                                      x = – 2     

The larger root:    – 1 is larger than – 2.     

The criteria c > a > 0 and a – b + c = 0 also work for  x² + 4x + 3   (x + 1)(x + 3)     

                                                                             and  x² + 5x + 4   (x + 1)(x + 4)     

                                                                             and  x² + 6x + 5   (x + 1)(x + 5)     

                                                                             and  x² + 7x + 6   (x + 1)(x + 6)     

                                                                             and you can keep going as long as you want to.     

_.     

Triangle DEF is an equilateral triangle with a side length of 3.     

A point P is chosen at random within triangle DEF.     

What is the probability that the length DP is at most 1?      

Draw an equilateral triangle DEF, side 3.     

Draw a circle, center at D, 1 inch radius.     

DP will be < 1 if it falls in or on the sector of the circle     

that's inside the triangle.     

Area of entire circle  =  (3.1416) • 1²  =  3.1416     

Area of sector inside triangle  =  (60^o / 360^o) • 3.1416  =  0.5236     

Area of triangle (per formula)  =  [ sqrt(3) / 4 ] • 3²  =  3.8971     

Probability  =  0.5236 / 3.8971  =  0.1344  ——>  as a percent  =  13.44%     

_.     

Twice the sum of a number and seven is three times the number.     

   2 • (x + 7)  =  3x     

       2x + 14  =  3x     

                –x  =  –14     

                  x  =  14     

_.     

Yes, it converges

there are 25 ways including order. only 5 

1,1

1,2

1,3

2,1

3,1

don't work

the answer is (25-5)/25 which is 20/25 or 4/5, or as bosco said, 80%

so it converges right? alan?

is it doable without arccos?

is there a way to do it without arcos?

This means that these do not happen with 10 flips. 

Firstly, we calculate the probability that there are 3 heads in a row

Ways to have 3 heads in a row occur 8 times 1

7 times: 

6 times:

5 times:

4 times:

3 times:

2 times:

1 time:

find the amount of ways then multiply by 1/8

Then do the same for 4 tails in a row. Then substract it all from 1

First, we list the numbers each number can be next to

Note no number can be next to itself, and every number other than the ones on the side are next to two values

1: 5,6

2: 3,4,5,6

3: 2,4,5,6

4: 2,3,5,6

5: 1,2,3,4,6

6: 1,2,3,4,5

We realize that every number can be next to any value except 1, and only 1 cannot be next to most numbers. We have two cases

1. 1 is in the middle (not on the edge)

1 must be next to 5 and 6, so we make a group of 1,5,6, where 1 must be in the middle, which has 2 arrangements within. Outside there are 4! = 24 arrangements, making 24 x 2 = 48

2. 1 is on the edge

1 must be next to 5 or 6, so there are 2 possibilities for that. The 1 can also be on either edge, making another 2 ways. The other 4 have 4! = 24 ways, therefore there are 2 x 2 x 24 = 96

Therefore, the total ways is 48 + 96 = ... 

I'm going to let you do the addition yourself

Just kidding it is 144

firstly, you realize that \(a_n = 1+1/n(n+1)\), which is \((n^2+n+1)/n^2+n\) No idea what to do next, just a idea!

Ms. Montgomery invested $800 at 8% simple interest at the beginning of each year for a period of ten years. Find the total accumulated value of all the investments at the end of the 10-year period.     

Beginning year balance           End of year balance     

  1st yr               800.00            1.08 •   800.00 =    864.00     

  2nd yr              864.00            1.08 •   864.00 =    933.12     

  3rd yr               933.12            1.08 •   933.12 =  1007.77     

  4th yr             1007.77            1.08 • 1007.77 =  1088.39     

  5th yr             1088.39            1.08 • 1088.39 =  1175.46     

  6th yr             1175.46            1.08 • 1175.46 =  1269.50     

  7th yr             1269.50            1.08 • 1269.50 =  1371.06     

  8th yr             1371.06            1.08 • 1371.06 =  1480.74     

  9th yr             1480.74            1.08 • 1480.74 =  1599.20     

10th yr             1599.20            1.08 • 1599.20 =  1727.14     

Another way to calculate the final balance would be 800 • 1.08¹⁰  =  1727.14     

_.     

state the value of c so it is a perfect square trinomial. the equation is 4x^2-24x+c.     

Figure out the factors.  Since it's a perfect square, both factors will be the same.     

The first term will be sqrt(4x²).  So first we have           (2x      )(2x      )     

The coefficient of the x¹ term is negative     

so the sign in the middle will be negative                      (2x –   )(2x –   )     

To determine the constant, you have to figure      

2x times what, plus 2x times what, equals 24x      

With a little thought, you realize it has to be 6.              (2x – 6)(2x – 6)     

Multiplied out, it is 4x² – 24x + 36, so c = 36     

_.     

 The Gonzalez family has a combined monthly income of $1850. Their food budget is $ 380 per month.     

 What percent of their monthly income is budgeted for food? (Round your answer to the nearest whole percent.)     

 Percent is 100 times the part of the whole divided by the whole.     

 %  =  100 • 380 / 1850     

       =  100 • 0.2054     

       =  20.54    ——>>  rounded as per problem is 21%     

_.     

What is the average speed (in feet/second) of a baseball that is thrown 88 feet in 4 seconds?     

(Write your answer to the nearest hundredth, or second decimal place.)     

Velocity  =  Distance / Time     

              =  88 ft / 4 sec     

              =  22 ^ft/_sec     

_.     

The area of one lateral face of a right pyramid with an equilateral triangular base is 75 square meters. If the slant height is 25 meters, what is the length of the side of its base, in meters?     

The slanted face is a triangle.  A_triangle  =  (1/2)(base)(height)     

                                                          75  =  (1/2)(base)(25)     

                                                   75 / 25  =  (1/2)base     

                                                            3  =  (1/2)base     

                                                      base  =  6     

_.     

 Two containers designed to hold water are side by side, both in the shape of a cylinder.     

 Container A has a diameter of 30 feet and a height of 18 feet.     

 Container B has a diameter of 26 feet and a height of 19 feet.     

 Container A is full of water and the water is pumped into Container B until Conainter B is completely full.     
 After the pumping is complete, what is the volume of the empty space inside Container A, to the nearest tenth of a cubic foot?      

V_CylB  =  π r² h     

V_CylB  =  (3.1416)(13²)(19)     

V_CylB  =  10,087.6776     

V_CylB  =  10,087.7 ft³    ——>>  Full container B was pumped out of container A.     

                                                   The amount that container B holds is how much     

                                                    empty space is left in container A.     

_.     

  A bouquet of lilies and tulips has 20 flowers. Lilies cost $3 each and tulips cost $2 each.     

  The bouquet costs $48. How many lilies x and tulips y are in the bouquet?      

  3L + 2T  =  48                      ——>   3L + 2T  =  48     

    L +   T  =  20   multiply x2  ——>   2L + 2T  =  40     

                             subtract     ——>             L  =   8 Lilies     

    L  +  T  =  20     

            T  =  20 – 8                ——>             T  =  12 Tulips     

_.