Circular Plate Paint Scaling Calculation

The amount of paint needed is directly proportional to the surface area of the plate to be painted. Since the plates are circular, their area is calculated using the formula for the area of a circle, which is $A = \pi r^2$, where '$r$' is the radius.

Relating Paint Volume to Area

Let $P_1$ be the paint required for the first plate with radius $r_1$, and $P_2$ be the paint required for the second plate with radius $r_2$. The areas are $A_1 = \pi r_1^2$ and $A_2 = \pi r_2^2$.

The ratio of the paint required is equal to the ratio of the areas:

$ \frac{P_2}{P_1} = \frac{A_2}{A_1} = \frac{\pi r_2^2}{\pi r_1^2} = \frac{r_2^2}{r_1^2} $

Calculating Paint for the Larger Plate

We are given:

• Paint for the first plate, $P_1 = 30$ ml
• Radius of the first plate, $r_1 = 20$ cm
• Radius of the second plate, $r_2 = 80$ cm

Substitute these values into the ratio formula:

$ \frac{P_2}{30 \text{ ml}} = \frac{(80 \text{ cm})^2}{(20 \text{ cm})^2} $

Calculate the squares of the radii:

$ \frac{P_2}{30 \text{ ml}} = \frac{6400 \text{ cm}^2}{400 \text{ cm}^2} $

Simplify the ratio of radii squared:

$ \frac{P_2}{30 \text{ ml}} = 16 $

Now, solve for $P_2$ by multiplying both sides by 30 ml:

$ P_2 = 16 \times 30 \text{ ml} $

$ P_2 = 480 \text{ ml} $

Therefore, 480 ml of paint is required for the larger plate.