Briquette Pieces Fec97C

1. **State the problem:** We need to find how many pieces of briquette are required to provide 80 MJ of energy, given the briquette's shape and energy density. 2. **Given data:** - Diameter of the outer cylinder $D = 12$ cm - Diameter of each hole $d = 2$ cm - Height of the cylinder $h = 9$ cm - Energy released per $1\ \text{cm}^3$ of briquette = $48$ kJ - Total energy required = $80$ MJ = $80,000$ kJ 3. **Calculate the volume of the briquette piece:** The briquette is a cylinder with multiple cylindrical holes. - Volume of outer cylinder: $$V_{outer} = \pi \left(\frac{D}{2}\right)^2 h = \pi \left(\frac{12}{2}\right)^2 \times 9 = \pi \times 6^2 \times 9 = 324\pi\ \text{cm}^3$$ - Volume of one hole: $$V_{hole} = \pi \left(\frac{d}{2}\right)^2 h = \pi \left(\frac{2}{2}\right)^2 \times 9 = \pi \times 1^2 \times 9 = 9\pi\ \text{cm}^3$$ - Number of holes is not explicitly given, but from the diagram description, assume there are $n$ holes. Since the problem does not specify the number of holes, we assume the number of holes is such that the total volume of holes is subtracted from the outer volume. 4. **Calculate the effective volume of briquette:** $$V_{briquette} = V_{outer} - n \times V_{hole} = 324\pi - 9\pi n = \pi (324 - 9n)\ \text{cm}^3$$ 5. **Calculate the energy per piece:** $$E_{piece} = V_{briquette} \times 48 = 48 \pi (324 - 9n)\ \text{kJ}$$ 6. **Calculate the number of pieces required:** $$N = \frac{80,000}{E_{piece}} = \frac{80,000}{48 \pi (324 - 9n)}$$ 7. **Simplify the expression:** $$N = \frac{80,000}{48 \pi (324 - 9n)} = \frac{80,000}{48 \pi \times 9 (36 - n)} = \frac{80,000}{432 \pi (36 - n)}$$ 8. **Final formula for number of pieces:** $$\boxed{N = \frac{80,000}{432 \pi (36 - n)}}$$ **Note:** To find a numerical value, the number of holes $n$ must be known from the problem or diagram. Since the number of holes is not given, this is the general formula for the number of pieces required.