The density of a solid spherical ball varies directly as its mass & inversely as the cube of the radius. When the mass of the ball is 500g & the radius is 5cm, its density is 2g per cm3. Calculate radius of a solid spherical ball of mass 340g & density of 10g per cm3.

The density $\rho$ of a solid spherical ball varies directly as its mass $m$ and inversely as the cube of its radius $r$. This can be written as:

$$\rho = \frac{k \cdot m}{r^3}$$

where $k$ is the constant of proportionality.

We are given the initial conditions: $m = 500$ g, $r = 5$ cm, and $\rho = 2$ g/cm³. Substitute these values into the equation:

$$2 = \frac{k \cdot 500}{5^3}$$ $$2 = \frac{k \cdot 500}{125}$$ $$2 = 4k$$

Divide by 4 to solve for $k$:

$$k = \frac{2}{4}$$ $$k = 0.5$$

We need to find the radius $r$ when the mass $m = 340$ g and the density $\rho = 10$ g/cm³. Use the calculated value of $k = 0.5$:

$$10 = \frac{0.5 \cdot 340}{r^3}$$ $$10 = \frac{170}{r^3}$$

Multiply both sides by $r^3$ and divide by 10 to solve for $r^3$:

$$r^3 = \frac{170}{10}$$ $$r^3 = 17$$

Take the cube root of both sides to find $r$:

$$r = \sqrt[3]{17}$$ $$r \approx 2.571$$

The radius of the solid spherical ball is approximately 2.57 cm.

$$2.57 \text{ cm}$$