Step 1: Find the radius of the sphere

Given the diameter $d = 84\text{ m}$, the radius $r$ is half of the diameter.

$r = \frac{d}{2} = \frac{84}{2} = 42\text{ m}$

Step 2: State the formula for the volume of a sphere

The volume $V$ of a sphere is given by the formula in terms of radius $r$.

$V = \frac{4}{3} \pi r^3$

Step 3: Substitute the values into the formula

Substitute $\pi = \frac{22}{7}$ and $r = 42\text{ m}$ into the volume expression.

$V = \frac{4}{3} \times \frac{22}{7} \times (42)^3$

Step 4: Calculate the final volume

Compute the value: $42^3 = 74088$. Dividing by $21$ gives $3528$, and multiplying by $4 \times 22 = 88$ yields $310,464\text{ m}^3$.

$V = \frac{4 \times 22 \times 42 \times 42 \times 42}{3 \times 7} = 310,464\text{ m}^3$