The jar shown is approximately a cylinder with a radius of 4 cm and a height of 16 cm. The jar is completely filled to the top with popcorn kernels. George determines the jar has a radius of AT MOST 9 kernels and a height of 20 kernels. Using the volume formula for a cylinder, AT MOST how many kernels would you estimate are in the jar?

To estimate how many kernels are in the jar, we will follow these steps:

1. **Volume of the Jar:**
   The volume \( V \) of a cylinder is given by the formula:
   \[
   V = \pi r^2 h
   \]
   where:
   - \( r = 4 \) cm (radius of the jar),
   - \( h = 16 \) cm (height of the jar),
   - \( \pi \approx 3.1416 \).

   Substituting the values:
   \[
   V = 3.1416 \times 4^2 \times 16
   \]
   \[
   V = 3.1416 \times 16 \times 16
   \]
   \[
   V \approx 3.1416 \times 256 = 804.25 \, \text{cm}^3
   \]

2. **Kernel Size:**
   George estimates that the jar has a radius of at most 9 kernels and a height of 20 kernels. Therefore:
   - The radius of one kernel can be approximated as \( \frac{4 \, \text{cm}}{9} \approx 0.44 \, \text{cm} \),
   - The height of one kernel can be approximated as \( \frac{16 \, \text{cm}}{20} = 0.8 \, \text{cm} \).

   Since each kernel can be roughly modeled as a cylinder, we can estimate the volume of one kernel using the formula for the volume of a cylinder. Let's assume the radius of each kernel is \( r_k = 0.44 \, \text{cm} \) and the height \( h_k = 0.8 \, \text{cm} \).

   \[
   V_k = \pi r_k^2 h_k
   \]
   \[
   V_k = 3.1416 \times (0.44)^2 \times 0.8
   \]
   \[
   V_k = 3.1416 \times 0.1936 \times 0.8 \approx 0.486 \, \text{cm}^3
   \]

3. **Total Number of Kernels:**
   To find the total number of kernels, divide the volume of the jar by the volume of one kernel:
   \[
   \text{Number of kernels} = \frac{804.25}{0.486} \approx 1655
   \]

Thus, George's estimate is that **there are at most 1,655 kernels** in the jar.

Would you like further details or have any questions?

---

Here are 5 related questions to explore:
1. What if the radius of the jar was 5 cm instead of 4 cm?
2. How would the number of kernels change if the height of the jar was 18 cm?
3. If the kernels were shaped like spheres instead of cylinders, how would that affect the estimate?
4. How many kernels would fit in the jar if their height was reduced by 10%?
5. What is the impact of using a different value for \( \pi \) (e.g., 3.14 or 3.14159)?

**Tip:** When estimating volumes, using different geometric shapes to model objects can lead to varying levels of accuracy.

Estimate how many popcorn kernels can fit in a cylindrical jar with a radius of 4 cm and height of 16 cm. This problem involves the geometry of cylinders and the concept of volume estimation, using the formula V = πr²h. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation