Answer: the proper diameter is 6.137 ft
Explanation:
first we find the volume of box using the relation, which is;
V = 2 x 5 x 1.5 = 15 ft³
we find the buoyant force on the box by calculating the weight of water displaced.
FB = V x y
where y is the specific weight of sea water(62.4 lbf/ft³)
so we Substitute
FB = 15 x 62.4 = 936 lbf
now we find the upward force required by the balloon
FR = (W - FB) x 120%
= 1.2 (W - FB)
where W is the weight of the box treasure(7000 lb)
so we Substitute,
FB = 1.2( 7000 - 936 ) = 7276.8 lbf
Because the universal gas constant contains a Rankine in its units, we make use of Rankine for our temperature
so we find the density of air at 3 atm using ideal gas relation,
Pair = p/RT
Here, p is the pressure acting (3 atm), R is the universal gas constant (1716 ft²/S²-R), and T is the temperature (520°R),
Substitute so we substitute
Pair = (3 * 2116.22) / (1516 * 520)
= 0.007114 lbf.ft³
next we find the specific weight of air;
Yair = Pair * g
g is acceleration due to gravity(32.2 ft/s²)
Yair =
0.007114 * 32.2
= 0.23 Ibf /ft³
Now we find diameter of the balloon by balancing the net force required
FR = (y - yair) * V - Wb
= (y - yair) x (π/6)d³ - Wb
d is the diameter of the balloon.
so we Substitute, 7276.8 lbf for FR,
62.4 Ibf/ft³ for y, 0.23 for lbf/ft³ for Yair, 350 lb for Wb
so
7276.8 = (62.4 - 0.23)πd³ - 250
d³ = 231.23 ft³
d = 6.137 ft
Therefore, the proper diameter is 6.137 ft
