A candle manufacturer sells cylindrical candles in sets of three. Each candle in the set is a different size. The smallest candle has a radius of 0.5 inches and a height of 3 inches. The other two candles are scaled versions of the smallest, with scale factors of 2 and 3. How much wax is needed to create one set of candles?
π * 0.5^2 * 3 (1+8+27) = 27π in^3
A candle manufacturer sells cylindrical candles in sets of three. Each candle in the set is a different size. The smallest candle has a radius of 0.5 inches and a height of 3 inches. The other two candles are scaled versions of the smallest, with scale factors of 2 and 3. How much wax is needed to create one set of candles?
To find out how much wax is needed to create one set of candles, we need to calculate the volume of each individual candle and then sum them up.
Let's start with the smallest candle, which has a radius of 0.5 inches and a height of 3 inches. The formula to calculate the volume of a cylinder is V = π * r^2 * h, where V is the volume, π (pi) is a mathematical constant approximately equal to 3.14159, r is the radius, and h is the height.
For the smallest candle:
V_1 = π * (0.5)^2 * 3
= π * 0.25 * 3
≈ 0.75π cubic inches
The next candle is a scaled version of the smallest candle, with a scale factor of 2. This means it will have twice the radius and height.
For the second candle:
V_2 = π * (2 * 0.5)^2 * (2 * 3)
= π * 1^2 * 6
= 6π cubic inches
The third candle is also a scaled version of the smallest candle, but with a scale factor of 3.
For the third candle:
V_3 = π * (3 * 0.5)^2 * (3 * 3)
= π * 1.5^2 * 9
≈ 20.25π cubic inches
Finally, to find out how much wax is needed to create one set of candles, we sum up the volumes of all three candles:
Total volume = V_1 + V_2 + V_3
= 0.75π + 6π + 20.25π
= 27π cubic inches
Therefore, to create one set of candles, approximately 27π cubic inches of wax is needed.
To calculate the amount of wax needed to create one set of candles, we need to calculate the volume of each candle and then sum them up.
The volume of a cylinder can be calculated using the formula: volume = π * radius^2 * height.
Let's calculate the volume of each candle:
1. Smallest candle:
radius = 0.5 inches
height = 3 inches
volume = π * 0.5^2 * 3
2. Second candle:
radius = 0.5 inches * 2 = 1 inch
height = 3 inches * 2 = 6 inches
volume = π * (1^2) * 6
3. Third candle:
radius = 0.5 inches * 3 = 1.5 inches
height = 3 inches * 3 = 9 inches
volume = π * (1.5^2) * 9
Now, we can calculate the total volume of wax needed for the set of candles by summing up the volumes of all three candles:
total volume = volume of smallest candle + volume of second candle + volume of third candle
total volume = π * 0.5^2 * 3 + π * (1^2) * 6 + π * (1.5^2) * 9
By substituting the values and calculating, you can find the total volume of wax needed to create one set of candles.