The value V of a diamond varies directly as the square of its mass m. If a diamond with a mass of 10 grams is worth GH¢200.00, find (a) the value of a diamond with a mass of 30 grams; (b) the mass of a diamond worth GH¢5,000.00.

Step 1: Establish the relationship between value $V$ and mass $m$, and find the constant of proportionality $k$.
The value $V$ varies directly as the square of its mass $m$, so we can write the relationship as:

$$V = km^2$$

We are given that $V = \text{GH}\cent 200.00$ when $m = 10$ grams. Substitute these values to find $k$:

$$200 = k(10)^2$$ $$200 = 100k$$ $$k = \frac{200}{100}$$ $$k = 2$$

So, the relationship is $V = 2m^2$.

a) Step 2: Find the value of a diamond with a mass of 30 grams.
Using the relationship $V = 2m^2$, substitute $m = 30$ grams:

$$V = 2(30)^2$$ $$V = 2(900)$$ $$V = 1800$$

The value of the diamond is GH$\cent$1,800.00.

$$\boxed{\text{GH}\cent 1,800.00}$$

b) Step 3: Find the mass of a diamond worth GH$\cent$5,000.00.
Using the relationship $V = 2m^2$, substitute $V = 5000$:

$$5000 = 2m^2$$

Divide both sides by 2:

$$m^2 = \frac{5000}{2}$$ $$m^2 = 2500$$

Take the square root of both sides:

$$m = \sqrt{2500}$$ $$m = 50$$

The mass of the diamond is 50 grams.

$$\boxed{50 \text{ grams}}$$