If the balloon is subjected to a net uplift force of F 800

If the balloon is subjected to a net uplift force of F = 800 N, determine the tension developed in ropes AB, AC, AD. The term tension means force.

Solution

Ok here\'s how to solve, hopefully U don\'t mind doing the math :>)

Let the intersection of the black vertical line extending from pt A to the ground = pt O
Using the physical dimensions given:
calculate the three angles: BAO, CAO and DAO
example for BAO:
distance BO = 1.5²+2² = 6.25 = 2.5
tan (BAO) = BO/AO = 2.5/6
arctan 2.5/6 = BAO = 22.6°
etc for angles CAO and DAO

Once all three angles have been obtained:
VERTICAL component of tension in all three tether lines AB , AC and AD = 800 N
so if the tension in these lines is: Tab, Tac, and Tad then write:
Tab(cos 22.9°) + Tac(Cos CAO) + tad(cos DAO) = 800

A 2nd equation is obtained from the fact that the HORIZONTAL components of the tension in all three tether lines must be in equilibrium. This means that
Tab(sin 22.9°) + Tac(sin CAO) + Tad(sin DAO) = 0

Finally notice that the attachment of AD is on the y-axis but the attachments of AB and AC are offset and on either side of the y-axis. This means that the force x-components of OB and OC must balance each other.
Tab(sin 22.9°)[1.5²/1.5²+2²] = Tac(sin CAO)[2²/2²+3²]
(0.14)Tab = (0.308)Tac(sin CAO)

With these three equations U should be able to solve for all three tensions: