CPhill

AD = CF    (1)

BD = DF     

Using (1).....this must mean that

(AB + BD)  = (CD + DF)    (3)

Since BD = DF....then this implies that

(AB + BD)  =  (CD + BD)     subtract  BD  from both sides

AB   =   CD

I don't think we  need the Law of Sines or Cosines, here

Call home plate  point A

Call third base point B

Call the point  at the right field fence  C

We have a right triangle  ABC  with angle BAC  =  90°

The distance we are  looking for is the hypotenuse= BC

AC  is one leg  = 300

AB is the other leg  =  90 ft from home plate to third base

So

BC  = √ [ AC^2 + AB^2]  = √  [ 300^2 + 90^2]  = √ 98100  ≈  313.2 ft

10x - 3y           3

_______  =   ___     cross-multiply

13x - 2y           5

5 [ 10x - 3y ]  =   3 [ 13x - 2y ]   simplify

50x - 15y =  39x  - 6y        add  15y to both sides.....subtract  39x  from both sides

11x  = 9y

x / y   =  9 / 11

x : y  = 9 /11

The total number of  votes   must have been 60  / .24   =     250

To win....a candidate must get  more than  250/2 =  125  votes

Note that 126 votes would do the trick !!!!

The  hedgehog  originally got  .42 * 250   = 105 votes

So....out of the 60 voters.....he  (or she) needs to get at least    126 - 105  =    21 votes from them to win

5x + 12y  = 13

12y  = 13 - 5x

y  =  [ 13 - 5x] / 12

x^2 + y^2   =

x^2  + [  (13 -5x ) /12  ]^2

x^2  + (1/144) [ 13 - 5x ]^2

Take the derivative of this and set to 0

2x  +  (2/144) [ 13 - 5x ] (-5)  =  0      simplify

2x - (10/144) [ 13 - 5x ]  = 0

2x - (5/72] [ 13 - 5x ]  =  0

2x   =   (5/72)[ 13 -5x ]

(144/5)x  =  13 - 5x

(144/5)x + 5x  = 13

[ (144 + 25 )/ 5 ] x  = 13

[169/5] x  = 13

x = [ 13 * 5 ] / 169

x =  5/13

y = [ 13 - 5 ( 5/13) ] / 12   =   [ 13 - 25/13] / 12   =  [ 169 - 25] / 156  =  144 / 156  =  12/13

So.....

min  [ x^2 + y^2 ]    =   

[5/13]^2  + [12/13]^2     =   

[ 5^2 + 12^2 ] / 13^2  = 

 [ 25 + 144 ] / 169  = 

169/169  = 

1

Very nice solution, Melody   !!!!

No prob, TacoBell

We just need to  realize  how to find angle  x

However.....we  don't need the sine here.......the sine is the opposite / hypotenuse

But we  actually have    opp / adj   =  tangent

So  we need

tan x   =  8/12    to  find  this angle we need to  take the arctangent   [ also called the inverse tangent]

I don't know how it might be noted on your calculator......most  use  tan^-1

[ Make sure your calculator is in degree mode  !!!]

So  we have

tan ^-1  ( 8/12)  =  x

33.69°   =  x    =    33.7° [ to one decimal place ]

Remember that to find an angle, we need to use the inverse functions  ]

Let me know if you have any problems here

Last one

We have 30 values

The median  occurs between th 15th and 16th value

This  is in the interval   20 ≤ h < 30

Second one

a)   medan

We have 30 values.....the median  falls between the 15th and 16th value

This = 3 goals

b)  mean

[ 2(0) + 8(1) + 4(2)  + 10(3)  + 6(4) ]  / 30  =  70 / 30  =  7 / 3   =  2 + 1/3   goals

First one

b)    Note  that  3 numbers are contained in set A    and  7  are not in set A

So.....A '  means the elements  not in set A

So  the probability that a number is not in set A  =      7  / [ 3 + 7 ]  =   7 / 10  =   .7