A.     Stephanie puts 30 cubes in a box

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A.     Stephanie puts 30 cubes in a box. The cubes are (1)/(2) inch on each side. The box holds 2 layers with 15 cubes in each layer. What is the volume of the box?

B.     Evaluate . First find the sum as a function of ; see Exercise 2.1.4b.

The correct answer and explanation is:

A. Volume of the Box

Given:

  • 30 cubes are placed in the box.
  • Each cube has a side length of 12\frac{1}{2} inch.
  • The box holds 2 layers, with 15 cubes in each layer.

To find the volume of the box:

The volume of the box can be calculated by determining its dimensions: length, width, and height.

  1. Volume of one cube: The formula for the volume of a cube is: Vcube=side length3V_{\text{cube}} = \text{side length}^3 For one cube with a side length of 12\frac{1}{2} inch: Vcube=(12)3=18 cubic inchesV_{\text{cube}} = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \, \text{cubic inches}
  2. Total volume of the 30 cubes: Since there are 30 cubes in the box, the total volume of the cubes is: Vtotal cubes=30×18=308=3.75 cubic inchesV_{\text{total cubes}} = 30 \times \frac{1}{8} = \frac{30}{8} = 3.75 \, \text{cubic inches}
  3. Dimensions of the box:
    • The box holds 2 layers, and each layer contains 15 cubes. Thus, the length and width of the box are based on the number of cubes in one layer.
    • Since the cubes are 12\frac{1}{2} inch on each side, the length and width of the box are:
    length=15×12=7.5 inches\text{length} = 15 \times \frac{1}{2} = 7.5 \, \text{inches} width=15×12=7.5 inches\text{width} = 15 \times \frac{1}{2} = 7.5 \, \text{inches}
    • The height of the box is the height of 2 layers of cubes, which is:
    height=2×12=1 inch\text{height} = 2 \times \frac{1}{2} = 1 \, \text{inch}
  4. Volume of the box: The volume of the box is calculated using the formula for the volume of a rectangular prism: Vbox=length×width×heightV_{\text{box}} = \text{length} \times \text{width} \times \text{height} Vbox=7.5×7.5×1=56.25 cubic inchesV_{\text{box}} = 7.5 \times 7.5 \times 1 = 56.25 \, \text{cubic inches}

Thus, the volume of the box is 56.25 cubic inches.

B. Evaluation of the Sum

It seems like the second part of your question refers to evaluating a sum as a function of something. However, the function you’re asking to evaluate is not fully specified in the text. If the sum is in reference to a series, the sum may be something like an arithmetic or geometric series.

Please clarify the sum you’re trying to evaluate, and I’ll provide a detailed explanation with the correct answer.

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