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The problem in the image seems to be about calculating the maximum number of cupcakes that can fit into a box. Here’s a breakdown of the information provided:

1. **Box Details:**
   - The box is shaped like a rectangular prism.
   - The width of the box is 8 cm.
   - The volume of the box is 720 cm³.
   - The box's length is half of the box's height.

2. **Cupcake Details:**
   - Each cupcake has a circular base with a diameter of 3.5 cm.

### To solve this problem, we can follow these steps:

1. **Calculate the Dimensions of the Box:**
   - Let the height of the box be \( h \) cm. Then, the length \( l \) is \( \frac{h}{2} \) cm.
   - We know the width \( w = 8 \) cm.
   - The volume formula for a rectangular box is \( V = l \times w \times h \).
   - Substituting the given values: 
   \[
   720 = \left(\frac{h}{2}\right) \times 8 \times h
   \]
   Simplifying the equation:
   \[
   720 = 4h^2
   \]
   \[
   h^2 = \frac{720}{4} = 180
   \]
   \[
   h = \sqrt{180} \approx 13.42 \text{ cm}
   \]
   - Therefore, the height \( h \) is approximately 13.42 cm, and the length \( l \) is \( \frac{13.42}{2} \approx 6.71 \) cm.

2. **Calculate the Number of Cupcakes that can Fit:**
   - Each cupcake occupies a circular area of \( \frac{\pi \times d^2}{4} \) where \( d = 3.5 \) cm, so the area for one cupcake is:
   \[
   A_{\text{cupcake}} = \frac{\pi \times 3.5^2}{4} \approx 9.62 \text{ cm}^2
   \]
   - The area of the bottom of the box (i.e., the base) is:
   \[
   A_{\text{base}} = l \times w = 6.71 \times 8 \approx 53.68 \text{ cm}^2
   \]
   - To find the number of cupcakes that can fit, we calculate:
   \[
   \text{Number of cupcakes in one layer} = \frac{A_{\text{base}}}{A_{\text{cupcake}}} \approx \frac{53.68}{9.62} \approx 5.58
   \]
   - Since we can only fit whole cupcakes, at most 5 cupcakes can fit on one layer.
   - The number of layers in the box is determined by the height. Assuming one cupcake fits in 3.5 cm height (equal to its diameter):
   \[
   \text{Number of layers} = \frac{h}{3.5} = \frac{13.42}{3.5} \approx 3.83
   \]
   - Again, only whole layers count, so you can have 3 layers.

   Finally, the total number of cupcakes is:
   \[
   \text{Total cupcakes} = 5 \times 3 = 15
   \]

So, the maximum number of cupcakes that can be fit into the box is **15 cupcakes**.

Would you like more details or have any questions? 

### Relative Questions:
1. How would the number of cupcakes change if the diameter was 4 cm instead of 3.5 cm?
2. How would the calculation differ if the box volume was 1000 cm³ instead of 720 cm³?
3. What if the box height was doubled, how many cupcakes could then fit?
4. How would the calculation change if the cupcakes were cylindrical with a height of 4 cm?
5. If the box's shape was a cylinder instead of a rectangular prism, how would you calculate the number of cupcakes?

### Tip:
Always make sure to check if your calculated dimensions are reasonable, especially when approximations are involved. This helps avoid errors in real-life applications.

Learn how to calculate the maximum number of cupcakes that can fit into a rectangular prism box. This problem involves geometry and volume calculations, focusing on the dimensions of the box and the area occupied by each cupcake. Suitable for Grades 7-9 students learning about geometry and volume calculations.

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