\text{Find the volume of a hemisphere, the circumference of whose base is}\mathrm{ 35.2}\text{cm (rounded off to two decimal places; use}\pi =\frac{22}{7}\text{).}

The correct option is 367.96 cm³.

To find the volume of a hemisphere, we first need to determine its radius. Let the radius of the hemisphere be denoted as r.

The base of a hemisphere is a circle, and the circumference of this base is given as 35.2 cm. The formula for the circumference of a circle is:

$$\text{Circumference}=2\pi r$$

Using $\pi =\frac{22}{7}$ and substituting the given circumference, we get:

$$35.2=2\times \frac{22}{7}\times r$$

Now, solve for the radius r:

$$35.2=\frac{44}{7}\times r$$

$$r=\frac{35.2\times 7}{44}$$

$$r=0.8\times 7$$

$$r=5.6\text{cm}$$

Now that we have the radius, we can calculate the volume of the hemisphere. The formula for the volume of a hemisphere is:

$$\text{Volume}=\frac{2}{3}\pi r^3$$

Substitute the values of $\pi$ and r into the formula:

$$\text{Volume}=\frac{2}{3}\times \frac{22}{7}\times {\left(5.6\right)}^3$$

$$\text{Volume}=\frac{2}{3}\times \frac{22}{7}\times 5.6\times 5.6\times 5.6$$

Simplify the terms by dividing 5.6 by 7:

$$\text{Volume}=\frac{2}{3}\times 22\times 0.8\times 5.6\times 5.6$$

First, calculate the product in the numerator:
$$2\times 22\times 0.8=35.2$$
$$5.6\times 5.6=31.36$$
$$35.2\times 31.36=1103.872$$

Now, divide by 3:

$$\text{Volume}=\frac{1103.872}{3}\approx \mathrm{367.95733...}$$

Rounding off to two decimal places, we get:
$$\text{Volume}\approx 367.96{\text{cm}}^3$$