The magnitudes of the forces that the ropes must exert on the knot connecting are :
- F₁ = 118 N 
- F₂ = 89.21 N 
- F₃ = 57.28 N 
<u>Given data :</u>
Mass ( M ) = 12 kg 
∅₂ = 63° 
∅₃ = 45°
<h3>Determine the magnitudes of the forces exerted by the ropes on the connecting knot</h3><h3 />
a) Force exerted by the first rope = weight of rope 
 ∴  F₁ = mg 
      = 12 * 9.81 ≈  118 kg 
<u>b) Force exerted by the second rope </u>
applying equilibrium condition of force in the vertical direction
F₂ sin∅₂ + F₃ sin∅₃ - mg = 0  ---- ( 1 )
where: F₃ = ( F₂ cos∅₂ / cos∅₃ ) --- ( 2 )  applying equilibrium condition of force in the horizontal direction 
Back to equation ( 1 ) 
F₂ =  [ ( mg / cos∅₂ ) / tan∅₂ + tan∅₃ ] 
    = [ ( 118 / cos 63° ) / ( tan 63° + tan 45° ) ]
    = 89.21 N 
<u />
<u>C ) </u><u>Force </u><u>exerted by the</u><u> third rope </u>
Applying equation ( 2 ) 
F₃ = ( F₂ cos∅₂ / cos∅₃ ) 
     = ( 89.21 * cos 63 / cos 45 )
     = 57.28 N 
Hence we can conclude that The magnitudes of the forces that the ropes must exert on the knot connecting are :
F₁ = 118 N, F₂ = 89.21 N, F₃ = 57.28 N 
Learn more about  static equilibrium : brainly.com/question/2952156