Keri has some ball bearings - Junior Cycle Mathematics - Question 3 - 2017

First, calculate the volume of the sphere with radius 25 mm using the same volume formula:

$$V = \frac{4}{3} \pi (25)^3 = \frac{4}{3} \pi (15625) = \frac{62500}{3} \pi ext{ mm}^3$$

Next, determine how many ball bearings are needed to match this volume:

$$\text{Number of ball bearings} = \frac{\frac{62500}{3} \pi}{288 \pi} = \frac{62500}{3 \times 288} \approx 72.3$$

Rounding up, Keri must melt down at least 73 ball bearings.

First, calculate the total volume from all 350 ball bearings:

$$\text{Total volume} = 350 \times 288 \pi = 100800 \pi ext{ mm}^3$$

Now use the volume formula to find the radius of the largest sphere:

$$\frac{4}{3} \pi r^3 = 100800 \pi$$

Dividing both sides by $\pi$ gives:

$$\frac{4}{3} r^3 = 100800$$ $$\Rightarrow r^3 = 100800 \times \frac{3}{4} = 75600$$

Taking the cube root:

$$R = \sqrt[3]{75600} \approx 42.0 ext{ mm} \text{ (to the nearest mm)}.$$

Therefore, the radius of the biggest sphere Keri could make is approximately 42 mm.