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:
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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.
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Cupcake Details:
- Each cupcake has a circular base with a diameter of 3.5 cm.
To solve this problem, we can follow these steps:
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Calculate the Dimensions of the Box:
- Let the height of the box be cm. Then, the length is cm.
- We know the width cm.
- The volume formula for a rectangular box is .
- Substituting the given values: Simplifying the equation:
- Therefore, the height is approximately 13.42 cm, and the length is cm.
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Calculate the Number of Cupcakes that can Fit:
- Each cupcake occupies a circular area of where cm, so the area for one cupcake is:
- The area of the bottom of the box (i.e., the base) is:
- To find the number of cupcakes that can fit, we calculate:
- 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):
- Again, only whole layers count, so you can have 3 layers.
Finally, the total number of cupcakes is:
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:
- How would the number of cupcakes change if the diameter was 4 cm instead of 3.5 cm?
- How would the calculation differ if the box volume was 1000 cm³ instead of 720 cm³?
- What if the box height was doubled, how many cupcakes could then fit?
- How would the calculation change if the cupcakes were cylindrical with a height of 4 cm?
- 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
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