You don’t give all the required information, but I’ll make some assumptions and you tell me where I’m wrong.
I’m going to assume the water is 3’ deep. Water weighs about 62 lbs. per cubic foot so the water in the tub will weigh a total of:
7’-10” x 9’-2” x (62 x 3’ deep) = 13,400#
I did not add weight for equipment because I assume there are cutouts for seats, which would reduce the amount of water (weight).
In addition, I did not add weight for occupants, because that is considered an impact load (load that is not on the structure for 7 days). This weight is calculated differently.
Assuming the hot tub sits across the 2x10’s in the 7’-10”direction, the 2x10’s at 12” on center spanning 7’ can support about 450 lbs. per foot depending on the species and grade of wood. The water and tub weighs about 3’ x 62 = 186 lbs. per foot. Adding 4 people each weighing about 200 lbs. each adds another 12 lbs. per square foot. Therefore, the total weight is:
Water = 186 psf (water 3’ deep) plus
Lumber = 8 psf (deck and joists) plus
People = 12 psf (4 people in tub) for a
Total = 206 psf (well within a safety factor)
However, in addition to supporting the tub, the double 2x12’s need to support a live load adjacent to the tub too. Therefore, the total load on the double 2x12’s is:
Tub/deck/4 people: 206 x 2.5 = 515 plf plus
Live load adjacent to tub,
which is 40 psf x 7’/2 = 140 plf plus
Lumber (deck, etc.) 8 psf x 7’/2 = 28 plf for a Total on double 2x12’s = 683 plf
If the 2-2x12’ are spanning 5’, then they’ll support about 1,800 lbs. per foot depending on the species and grade of wood. This is also well within a safety factor.
However, the total weight on the 4-4x6’s is about 13,400 lbs. / 4 posts = 3,350 lbs. plus about 1,700 lbs. live load for a total load of 5,050 lbs.
The 2x12’s are bolted to the 4x6’s in a double shear loading pattern. Assuming those are 5/8” bolts, they can only support about 510 lbs. each or about 1,020 lbs. at each 4x6 post.
The framing lumber will easily support the required loads, but the connection at the 4x6 posts is grossly inadequate. You’ll need to install a 2x6 vertical under each side of each 4x6 that is under the hot tub. Make sure it bears on the footing and fits tight under each 2x12.