Trig word problems please give full answer ASAP

Question

1.) 

a flagpole casts a 24 foot long shadown when the sun is at an angle of elevation of 65 degrees. how tall is the pole?

2.)

a painter needs to reach high up the side of a house to finish a painting job. At what angle with the ground should the painter place a 20 foot ladder so that the top of the ladder reaches exactly 18 feet up the side of the house? Round your answer to the nearest tenth.

3.)

a boat leaves point A traveling at 20 mph. At the same time, another boat leaves point A traveling at 14 mph in a heading that is 80 degrees from the first boat. After 2 hours, how far apart are the two boats?

Answer 1

(1)

 

    C |  

       |

       |

       |  

       |

       |                              65º

    B |______________________ A

                        24 ft

 

 

    The height of the pole is side BC.

    Angle A is 65º

 

    tangent A = opposite/adjacent

    tangent 65º = AB/24

    AB/24 = tangent 65º

    AB = (24)•tangent 65º

    AB = 24(2.14451)

    AB = 51.47 ft

 

    Flagpole is approximately 51.47 ft tall.

 

___________________________________________________________________________

 

(2)

 

    C |
       |
       |
       | 18 ft
       |
       | 
    B |_________ A

 

    We need to determine angle A.

    Side AC es the ladder, which is 20 ft.

    We need the ladder to be 18 feet up the side of the building (side BC)

 

    sine A = opposite/hypotenuse

              = 18/20

              = 9/10

              =.9

           A = sine^-1(.9)

           A = 64.16º

 

    Angle A is approximately 64.16 degrees.

 

_____________________________________________________________________________

 

(3)

 

     A ___________  B

        \                /

         \ 80º        /

          \            /

           \          /

            \        /

             \      /

              \    /

               \  /

                \/

                C

 

    We are looking for side BC.

    

    Since rate x time is distance, we can find the length of AB.  Rate is 20 mph and time is 2 hours

    (20)(2) = 40 ... side AB is 40

 

    Similarly, we can find the length of AC.  Rate is 14 mph and time is 2 hours.

    (14)(2) = 28 ... side AC is 28

 

    We can use the law of cosines to find side BC.

 

      a² = b² + c² - 2bc• cos A

          = (28)² + (40)² - 2(28)(40)•cos 80º

          = 784 + 1600 - 2240(.17365)

          = 2384 - 388.97

          = 1995.03

       a = √(1995.03)

          = 44.67

 

     Thus, the 2 boats are 44.67 miles apart.