Math Problem Statement

Solution

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 hh cm. Then, the length ll is h2\frac{h}{2} cm.
    • We know the width w=8w = 8 cm.
    • The volume formula for a rectangular box is V=l×w×hV = l \times w \times h.
    • Substituting the given values: 720=(h2)×8×h720 = \left(\frac{h}{2}\right) \times 8 \times h Simplifying the equation: 720=4h2720 = 4h^2 h2=7204=180h^2 = \frac{720}{4} = 180 h=18013.42 cmh = \sqrt{180} \approx 13.42 \text{ cm}
    • Therefore, the height hh is approximately 13.42 cm, and the length ll is 13.4226.71\frac{13.42}{2} \approx 6.71 cm.
  2. Calculate the Number of Cupcakes that can Fit:

    • Each cupcake occupies a circular area of π×d24\frac{\pi \times d^2}{4} where d=3.5d = 3.5 cm, so the area for one cupcake is: Acupcake=π×3.5249.62 cm2A_{\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: Abase=l×w=6.71×853.68 cm2A_{\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: Number of cupcakes in one layer=AbaseAcupcake53.689.625.58\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): Number of layers=h3.5=13.423.53.83\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: Total cupcakes=5×3=15\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.

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Math Problem Analysis

Mathematical Concepts

Geometry
Volume Calculation
Area Calculation

Formulas

Volume of a rectangular prism
Area of a circle

Theorems

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Suitable Grade Level

Grades 7-9