To determine how many candles Joel can make, we need to break the problem down into a series of steps involving key calculations.
1. Determine the radius of the mould:
Given:
- Diameter of the mould = 12 cm
Radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[
r = \frac{12}{2} = 6 \, \text{cm}
\][/tex]
2. Calculate the volume of the cylindrical mould:
The volume [tex]\( V \)[/tex] of a cylinder is given by the formula:
[tex]\[
V = \pi r^2 h
\][/tex]
where:
- [tex]\( r = 6 \, \text{cm} \)[/tex] (radius)
- [tex]\( h = 18 \, \text{cm} \)[/tex] (height)
Plugging in the values:
[tex]\[
V = \pi \times (6)^2 \times 18
\][/tex]
[tex]\[
V \approx 3.14159 \times 36 \times 18
\][/tex]
[tex]\[
V \approx 2035.75 \, \text{cm}^3
\][/tex]
3. Calculate the volume of melted wax needed per candle:
Joel fills each mould to [tex]\(\frac{7}{8}\)[/tex] of its height, so:
[tex]\[
\text{Effective height} = \frac{7}{8} \times 18 = 15.75 \, \text{cm}
\][/tex]
The effective volume of wax per candle [tex]\( V_{\text{candle}} \)[/tex]:
[tex]\[
V_{\text{candle}} = \pi \times (6)^2 \times 15.75
\][/tex]
[tex]\[
V_{\text{candle}} \approx \pi \times 36 \times 15.75
\][/tex]
[tex]\[
V_{\text{candle}} \approx 1781.28 \, \text{cm}^3
\][/tex]
4. Determine the total available volume of melted wax:
Given:
- 1 kg of solid wax makes [tex]\( 1170 \, \text{cm}^3 \)[/tex] of melted wax
- Total wax = 15 kg
Therefore, total volume of melted wax [tex]\( V_{\text{total}} \)[/tex]:
[tex]\[
V_{\text{total}} = 15 \times 1170
\][/tex]
[tex]\[
V_{\text{total}} = 17550 \, \text{cm}^3
\][/tex]
5. Calculate the maximum number of candles:
The number of candles [tex]\( N \)[/tex] is given by the total volume of melted wax divided by the volume of wax per candle:
[tex]\[
N = \frac{17550}{1781.28}
\][/tex]
[tex]\[
N \approx 9.85
\][/tex]
As Joel cannot make a fraction of a candle, he can make the largest whole number of candles, which is:
[tex]\[
N = 9
\][/tex]
Therefore, Joel can make 9 candles with the 15 kg of solid wax he has.
