To estimate the number of popcorn kernels in the jar, we can use the volume formula for a cylinder:

$V = \pi r^2 h$

where:

• $r$ is the radius,
• $h$ is the height, and
• $\pi$ is approximately 3.1416.

Step 1: Calculate the Volume of the Jar

The given jar has:

• Radius $r = 3 \, \text{cm}$
• Height $h = 18 \, \text{cm}$

Substituting into the formula:

$V_{\text{jar}} = \pi \times 3^2 \times 18 = \pi \times 9 \times 18 = 162\pi \, \text{cm}^3$

Approximating with $\pi \approx 3.1416$:

$V_{\text{jar}} = 162 \times 3.1416 \approx 508.938 \, \text{cm}^3$

So, the volume of the jar is approximately 508.94 cm³.

Step 2: Estimate the Volume of One Kernel

We know that George estimated:

• The radius of the jar is at most 11 kernels wide.
• The height of the jar is at most 20 kernels tall.

Assuming the kernels are roughly spherical in shape, we can estimate the diameter of one kernel. The diameter of 11 kernels equals the diameter of the jar (which is $6 \, \text{cm}$), so the diameter of one kernel is:

$\frac{6 \, \text{cm}}{11} \approx 0.545 \, \text{cm}$

Thus, the radius of one kernel is approximately:

$r_{\text{kernel}} \approx \frac{0.545}{2} = 0.2725 \, \text{cm}$

Now, the volume of one kernel, assuming it’s approximately spherical, can be calculated using the volume formula for a sphere:

$V_{\text{kernel}} = \frac{4}{3} \pi r_{\text{kernel}}^3$

Substituting the value of $r_{\text{kernel}}$:

$V_{\text{kernel}} = \frac{4}{3} \pi (0.2725)^3 \approx \frac{4}{3} \pi \times 0.0202 \approx 0.0846 \, \text{cm}^3$

Step 3: Estimate the Number of Kernels in the Jar

To estimate the number of kernels in the jar, we divide the volume of the jar by the volume of one kernel:

$\text{Number of kernels} = \frac{V_{\text{jar}}}{V_{\text{kernel}}} = \frac{508.94}{0.0846} \approx 6014$

Thus, at most 6,014 kernels could fit in the jar.

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Would you like more details on any part of this solution?

Here are 5 related questions to expand this:

1. How would the answer change if the kernels were not spherical but irregular in shape?
2. What assumptions were made in this problem?
3. How would using a different method to estimate kernel volume affect the result?
4. How would packing density affect the number of kernels that can fit?
5. How can we account for the gaps between kernels in real life?

Tip: For real-world packing problems, it’s useful to consider that spheres don’t pack perfectly; the efficiency is typically around 60-70% due to the empty spaces between them.