To solve the problem, let's break it into parts:

Step 1: Volume of the cylinder

The volume $V$ of a cylinder is given by: $V = \pi r^2 h$ Here, the base radius $r$ is $3x$ cm and the height is $h$ cm. Substituting $r = 3x$, we get: $V_{\text{cylinder}} = \pi (3x)^2 h = 9\pi x^2 h$

Step 2: Volume of one sphere

The volume of a sphere is given by: $V = \frac{4}{3} \pi r^3$ The radius $r$ of each sphere is given as $\frac{1}{2}x$. Substituting $r = \frac{1}{2}x$, we calculate the volume: $V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{1}{2}x\right)^3$ Simplify $\left(\frac{1}{2}x\right)^3$: $V_{\text{sphere}} = \frac{4}{3} \pi \cdot \frac{x^3}{8} = \frac{4\pi x^3}{24} = \frac{\pi x^3}{6}$

Step 3: Total volume of 270 spheres

Since 270 spheres are made, the total volume of all the spheres is: $V_{\text{total spheres}} = 270 \cdot V_{\text{sphere}} = 270 \cdot \frac{\pi x^3}{6}$ Simplify: $V_{\text{total spheres}} = 45 \pi x^3$

Step 4: Equating the volumes

The metal cylinder is melted to make the spheres, so the total volume of the cylinder equals the total volume of the spheres: $V_{\text{cylinder}} = V_{\text{total spheres}}$ Substitute the expressions: $9\pi x^2 h = 45 \pi x^3$

Step 5: Solve for $h$

Cancel $\pi$ on both sides: $9x^2 h = 45x^3$ Divide through by $9x^2$ (assuming $x \neq 0$): $h = \frac{45x^3}{9x^2} = 5x$

Final Answer:

$h = 5x$

Let me know if you'd like a deeper explanation or have further questions!

5 Related Questions:

1. How do you derive the formula for the volume of a cylinder?
2. What is the significance of equating the volumes in this type of problem?
3. How would the solution change if the number of spheres were different?
4. What happens if the radius of the spheres is increased while keeping the cylinder dimensions constant?
5. How can this concept be applied in real-life scenarios like recycling or manufacturing?

Tip:

Always double-check the units and dimensions when solving volume-related problems to ensure consistency and accuracy!