Brooke Martin
Estimating the number of peppermint candies that can fit inside a jar candle is a fun challenge that involves some interesting math. The problem is more complex than simply multiplying the dimensions of the jar by the volume of the candies, as the candies can be packed into the jar in various ways, depending on their shape and size. The shape of the jar also plays a role in determining the maximum number of candies that can fit inside. In general, the problem requires an understanding of packing density, which is the proportion of a container's volume that is filled by identical spheres or other shapes.
| Characteristics | Values |
|---|---|
| Volume of jar candle | 47 ounces |
| Volume of a peppermint candy | 0.09067 liquid ounces |
| Number of peppermint candies in a jar candle | 330 |
| Average number of contacts for a particle within any mixed container | 6 |
| Packing density of randomly packed identical spheres | 64% |
| Packing density of spheres with varying sizes | Smaller spheres have a higher packing density |
| Burn time of a 3 oz. peppermint-scented candle | 15 hours |
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What You'll Learn
Calculating the volume of the jar
To calculate the volume of the jar candle, we need to know its shape. The most common formula for volume is for a cylinder, which can be calculated using the formula: Volume = π • r² • h, where r is the radius of the base and h is the height of the cylinder.
If the jar is a cylinder, we can use the above formula to calculate its volume. First, we need to measure the radius of the jar candle's base. This can be done by measuring the diameter (the distance between any two points on the circumference of the circle) and dividing it by 2.
Once we have the radius, we can square it (multiply it by itself) and then multiply it by pi (approximately 3.14) and the height of the jar candle. This will give us the internal volume of the jar candle in cubic units (e.g. cubic inches or cubic centimetres).
If the jar candle is not a perfect cylinder, we can still estimate its volume by approximating it to a rectangle on each face and multiplying the internal height by the internal width by the internal depth. This will give us a reasonable approximation of the jar candle's volume.
Additionally, if the jar has a unique shape that is difficult to measure directly, we can fill it with water and then pour the water into a measuring device to determine the volume of the jar candle. This method assumes that the jar candle is watertight and does not leak.
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Calculating the volume of each candy
To calculate the volume of each candy, we need to make some assumptions about their shape. Let's assume that the candies are cylindrical, as this is a common shape for peppermint candies.
With this assumption, we can use the formula for the volume of a cylinder to calculate the volume of each candy. The formula for the volume of a cylinder is:
> Volume = pi x radius^2 x height
We need to measure or estimate the radius and height of one candy. Let's say the radius is 0.5 inches and the height is 1 inch (these values can be measured precisely with a ruler or estimated using a measuring tape).
Plugging these values into the formula, we get:
> Volume = pi x (0.5)^2 x 1
Calculating this expression gives us a volume of approximately 0.785 cubic inches for each candy.
Alternatively, we can approximate the candies as small spheres, which have a volume formula of:
> Volume = (4/3) x pi x radius^3
Using the same radius of 0.5 inches, we get:
> Volume = (4/3) x pi x (0.5)^3
Calculating this expression gives us a volume of approximately 0.52 cubic inches for each candy.
It's important to note that the shape of the candy may vary, and these calculations assume idealized shapes. The actual volume of each candy may differ slightly depending on its exact shape and any irregularities it may have.
By calculating the volume of each candy, we can now use the volume of the jar to estimate the total number of candies that can fit inside.
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Estimating the packing density
The next step is to estimate the volume of an individual peppermint candy. This can be approximated by assuming a cylindrical shape for the candies and using the formula: pi multiplied by the radius of the candy, squared, and then multiplied by the height of the candy. If the radius of the candy is 0.25 inches and the height is 1 inch, as provided in one example, the volume of each candy would be approximately 0.09067 cubic inches, or 0.09067 liquid ounces.
With these values, we can now estimate the packing density of the peppermint candies in the jar candle. One approach is to divide the internal volume of the jar candle by the volume of an individual candy. In the given example, this would yield a maximum count of 1410 candies (128 / 0.09067). However, this assumes a perfect packing efficiency, which is not typically achievable in practice.
To account for this, we can apply a packing density factor. Randomly packed identical spheres, such as spherical candies, tend to fill up approximately 64% of the volume of a container. Therefore, we can multiply our maximum count by 0.64 to estimate the achievable number of candies. In this case, 64% of 1410 candies would be 902 candies.
It is worth noting that the aspect ratio of the candies can also influence the packing density. If the candies are not perfectly spherical, their shape can affect how efficiently they pack together. In such cases, adjustments to the packing density factor may be necessary. For instance, if the candies have an aspect ratio of 0.5, the packing density may be slightly lower than 0.64, and a more conservative estimate could be used.
By following these steps and making appropriate assumptions about the jar candle's dimensions and the candies' shape and size, one can estimate the packing density and approximate the number of peppermint candies that can fit inside a jar candle.
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Considering the shape of the candy
When considering the shape of the candy, it is important to note that the answer to "how many peppermint candies fit in a jar candle?" will vary depending on the shape of the candies and how they are packed. If the candies are assumed to be sphercylinders, as one source suggests, then the aspect ratio and random packing density can be calculated to estimate the number of candies that can fit in the jar.
In the example provided, the candies are estimated to have an aspect ratio of 0.5, which gives a random packing density of 0.66 when averaged with the density of spheres. This means that the candies will fill up about 66% of the volume of the jar. By estimating the size of the jar and the size of the candies, the total number of candies that can fit inside can be calculated.
If the candies are assumed to be spherical, as in the experiments conducted by Brujic's team, the calculation may be simplified. Randomly packed identical spheres fill up about 64% of the volume of a given container. By taking 64% of the volume of the jar and dividing it by the volume of one candy, an estimate of the total number of candies that can fit inside can be obtained.
However, it is important to note that the shape of the candy can significantly impact the packing density and, therefore, the estimated number of candies that fit in the jar. For example, counting jelly beans would be more complicated than counting gumballs due to their irregular shape. Additionally, the size distribution of the candies can be tuned to maximize or minimize density, as suggested by Brujic.
In the case of peppermint candies, they often have a unique pillow shape, as mentioned by Atkinson Candy Co., or may be cylindrical or spherocylindrical as suggested by their packaging and appearance. Therefore, when considering the shape of the candy, it is important to take into account the specific dimensions and characteristics of the peppermint candies in question to accurately estimate the number of candies that can fit inside the jar candle.
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Using an average of guesses
Guessing the number of candies in a jar is a fun game, but it can be tricky to get an accurate answer. One method is to use an average of guesses, and there are a few ways to do this. Firstly, you can take the average of several people's guesses. If you have a group of people all guessing, add up all their estimates and divide by the number of guesses to get an average. This can be a good approach, especially if the jar is transparent or partially transparent, as people can use their eyes to estimate.
Another method, as suggested by a user on Reddit, is to create a replica setup with the same jar and candies. You can fill the jar with candies about ten times, counting how many fit in the jar each time, and then take the average of these ten results. This method will give you a good idea of the average packing density of the candies in the jar.
If you want to get more mathematical, you can calculate the internal volume of the jar by multiplying the internal height, width, and depth. Then, you can estimate the volume of an individual candy, either by measuring or using an approximation. For example, if the candies are cylindrical, you can use the formula pi x radius of the sweet x radius of the sweet x height of the sweet. Once you have the volume of one candy, you can divide the total volume of the jar by the volume of one candy to get the average number of candies that would fit in the jar.
It is worth noting that this calculation assumes that the candies are all the same size and shape and that there is no space between them. In reality, candies may vary in size and shape, which would affect how they pack together and the total number that fits in the jar. Researchers have found that randomly packed identical spheres fill up about 64% of the volume in a given container. So, if the candies are all the same size, you can take 64% of the volume of the jar and divide it by the volume of one candy to get an estimate of how many would fit.
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Frequently asked questions
This depends on the size of the jar candle and the candies. If you're using a 3 oz jar candle, you won't be able to fit many candies inside. However, if you have a larger jar candle, you can estimate the number of candies by calculating the internal volume of the jar and the volume of an individual candy.
Measure the internal height, width, and depth of the jar. Then, multiply these three numbers together to get the internal volume.
Measure the radius and height of a single candy. If the candies are cylindrical, use the formula pi x radius of the candy x radius of the candy x height of the candy. If the candies are spherical, simply use the formula pi x radius of the candy x radius of the candy x 4/3.
Yes, if the candies are all the same size, you can take 64% of the volume of the jar and divide it by the volume of one candy. This will give you a rough estimate of the number of candies that can fit inside the jar.
