Explanation:
the volume of a pyramid is
base area × inner height / 3
we have the base area : 7×7 = 49 in²
so, we need to get the inner height h.
fit this we need to get the outer height l.
the surface area is
base area (49 in²) + 4×side triangle area = 185 in².
one of these 4 side triangle area is
baseline (7 in) × l / 2
so, we have a surface area
49 + 4×7×l/2 = 185
49 + 2×7×l = 185
49 + 14×l = 185
14×l = 136
l = 136/14 = 68/7 = 9.714285714... in
h we get now via Pythagoras
c² = a² + b²
with c being the Hypotenuse (side opposite of the 90° angle), a and b being the legs.
l² = h² + 3.5²
94.36734694... = h² + 12.25
h² = 94.36734694... - 12.25 = 82.11734694...
h = 9.061862222... in
so, the volume of the pyramid is
49 × 9.061862222... / 3 = 148.0104163... in³ ≈
≈ 148.01 in³