Balloon problem

Zhowers

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Joined
Apr 17, 2019
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So halfway through the problem I noticed that the problem could have been used as a law of cosines, and I feel like I might be wrong so far. I'm pretty stuck and would appreciate any help! Thank you.
 

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HallsofIvy

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Jan 27, 2012
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Let the tension in the left rope be \(\displaystyle T_1\) and the tension in the right rope be \(\displaystyle T_2\). Draw a line straight down from the balloon. That divides the problem into two right triangles. On the left we have a hypotenuse of "length" \(\displaystyle T_1\) and angle 60 degrees. The vertical component of force is \(\displaystyle T_1 sin(60)\), downward, and horizontal component of force is \(\displaystyle T_1 cos(60)\), to the left. On the right we have a hypotenuse of "length" \(\displaystyle T_2\) and angle 25 degrees. The vertical component of force is \(\displaystyle T_2 sin(25)\), downward, and horizontal component of force is \(\displaystyle T_2 cos(25)\), to the right.

The total horizontal force must be 0: \(\displaystyle T_1cos(60)= T_2cos(25)\). The total vertical force is also 0 but that exerted by the ropes must offset the upward lifting force, 570 pounds: \(\displaystyle T_1sin(60)+ T_2sin(25)= 570\). Solve those two linear equations for \(\displaystyle T_1\) and \(\displaystyle T_2\).
 
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