The jar shown is approximately a cylinder with a radius of 3 cm and a height of 18 cm. The jar is completely filled to the top with popcorn kernels. George determines the jar has a radius of AT LEAST 7 kernels and a height of 23 kernels. Using the volume formula for a cylinder, AT LEAST how many kernels would you estimate are in the jar?

We can solve this problem by comparing the volume of the jar to the volume occupied by the kernels.

### Step 1: Volume of the jar
The volume \( V \) of a cylinder is given by the formula:

\[
V = \pi r^2 h
\]

where:
- \( r \) is the radius of the cylinder (jar),
- \( h \) is the height of the cylinder (jar).

Given:
- Radius \( r = 3 \, \text{cm} \),
- Height \( h = 18 \, \text{cm} \),

The volume of the jar is:

\[
V_{\text{jar}} = \pi (3)^2 (18) = \pi \times 9 \times 18 = 162\pi \, \text{cm}^3
\]
Thus, the volume of the jar is approximately:

\[
V_{\text{jar}} \approx 162 \times 3.1416 = 508.94 \, \text{cm}^3
\]

### Step 2: Volume occupied by a single kernel
George determines that the radius of the jar is approximately 7 kernels and the height is about 23 kernels. This means that the dimensions of each kernel can be estimated as follows:
- The radius of the jar is 7 kernels, so the radius of each kernel is approximately \( \frac{3}{7} \, \text{cm} \).
- The height of the jar is 23 kernels, so the height of each kernel is approximately \( \frac{18}{23} \, \text{cm} \).

Now, we estimate the volume of a single kernel, approximating it as a small cylinder:
- The radius \( r_{\text{kernel}} \approx \frac{3}{7} \, \text{cm} \),
- The height \( h_{\text{kernel}} \approx \frac{18}{23} \, \text{cm} \).

The volume of a single kernel is:

\[
V_{\text{kernel}} = \pi \left( \frac{3}{7} \right)^2 \left( \frac{18}{23} \right) = \pi \times \frac{9}{49} \times \frac{18}{23}
\]

\[
V_{\text{kernel}} = \pi \times \frac{162}{1127} \approx 0.451 \, \text{cm}^3
\]

### Step 3: Number of kernels in the jar
To estimate the total number of kernels, divide the volume of the jar by the volume of a single kernel:

\[
\text{Number of kernels} = \frac{V_{\text{jar}}}{V_{\text{kernel}}} \approx \frac{508.94}{0.451} \approx 1128
\]

Thus, the jar contains approximately **1128 kernels**.

Would you like further details or have any questions?

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Here are five related questions to expand on this topic:

1. How would the estimate change if the kernels were packed more loosely or more tightly?
2. How would you calculate the volume if the jar had a different shape (e.g., a cone)?
3. What other methods could be used to estimate the number of kernels in the jar without using volume?
4. How would the total number of kernels change if the jar's height were doubled?
5. How does the assumption that kernels are cylindrical affect the accuracy of the estimate?

**Tip**: Understanding how to approximate real-world objects with geometric shapes can help simplify complex volume problems.

Estimate how many popcorn kernels fit in a jar by comparing the jar's volume to the volume of each kernel. Learn how to use the formula for the volume of a cylinder and apply it to solve real-world problems. Suitable for students in Grades 9-12.

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