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Magazine Chapter 11: Problem 28
The specifications of a cardboard box indicated that it was to have the same volume as a rectangular box 6 inches by 14 inches by 20 inches, but with less surface area. Which size box would NOT meet these requirements? A 7 in. by 10 in. by 24 in. B 7 in. by 12 in. by 20 in. C 10 in. by 12 in. by 14 in. D 5 in. by 16 in. by 21 in.
Short Answer
Expert verified
Option D does not meet the requirements because it has a larger surface area.
Step by step solution
01
Calculate the Volume of the Original Box
We need the volume of the initial box to compare with other options. Calculate the volume using the formula for the volume of a rectangular box: \( V = L \times W \times H \).The sides are given as 6 inches, 14 inches, and 20 inches.\[ V = 6 \times 14 \times 20 = 1680 \text{ cubic inches} \]
02
Calculate the Surface Area of the Original Box
Now, calculate the surface area of the original box using the formula for the surface area of a rectangular box: \( A = 2(LW + LH + WH) \).\[ A = 2(6 \times 14 + 6 \times 20 + 14 \times 20) = 2(84 + 120 + 280) = 968 \text{ square inches} \]
03
Calculate Volume and Surface Area for Option A
For the box measuring 7 inches by 10 inches by 24 inches:Calculate the volume:\[ V = 7 \times 10 \times 24 = 1680 \text{ cubic inches} \] (volume matches the original)Calculate the surface area:\[ A = 2(7 \times 10 + 7 \times 24 + 10 \times 24) = 2(70 + 168 + 240) = 956 \text{ square inches} \]
04
Calculate Volume and Surface Area for Option B
For the box measuring 7 inches by 12 inches by 20 inches:Calculate the volume:\[ V = 7 \times 12 \times 20 = 1680 \text{ cubic inches} \] (volume matches the original)Calculate the surface area:\[ A = 2(7 \times 12 + 7 \times 20 + 12 \times 20) = 2(84 + 140 + 240) = 928 \text{ square inches} \]
05
Calculate Volume and Surface Area for Option C
For the box measuring 10 inches by 12 inches by 14 inches:Calculate the volume:\[ V = 10 \times 12 \times 14 = 1680 \text{ cubic inches} \] (volume matches the original)Calculate the surface area:\[ A = 2(10 \times 12 + 10 \times 14 + 12 \times 14) = 2(120 + 140 + 168) = 856 \text{ square inches} \]
06
Calculate Volume and Surface Area for Option D
For the box measuring 5 inches by 16 inches by 21 inches:Calculate the volume:\[ V = 5 \times 16 \times 21 = 1680 \text{ cubic inches} \] (volume matches the original)Calculate the surface area:\[ A = 2(5 \times 16 + 5 \times 21 + 16 \times 21) = 2(80 + 105 + 336) = 1042 \text{ square inches} \]This surface area is greater than the original box's 968 square inches.
07
Identify the Option That Doesn't Meet Requirements
After calculating all the options, we need to determine which box does not have a reduced surface area compared to the original 968 square inches.
Option D has a surface area of 1042 square inches, which is greater than 968 square inches, thus failing the requirement.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rectangular Box
A rectangular box, often referred to as a rectangular prism, is a three-dimensional shape. It has six faces, all of which are rectangles. Each box can be defined by its length (
), width (
), and height (
), typically labeled as _L_, _W_, and _H_. These dimensions determine the overall space it occupies and the exterior area which covers it. Understanding these dimensions is crucial when calculating both the volume and the surface area of the box.
- Length ( ): The longest side of the rectangle base.
- Width ( ): The shorter side of the base.
- Height ( ): The side perpendicular to the base.
Volume Calculation
The volume of a rectangular box refers to the total space within. It's often expressed in cubic units, like cubic inches or cubic centimeters. Calculating the volume involves multiplying the three dimensions of the box:\[ V = L \times W \times H \]For example, consider a box with dimensions 6 inches by 14 inches by 20 inches. Its volume can be calculated as follows:\[ V = 6 \times 14 \times 20 = 1680 \text{ cubic inches} \]Thus, the volume ensures us about the capacity of the box to hold material or items. This concept is especially important in shipping and storage contexts where capacity needs to be optimized.
Surface Area Reduction
Surface area measures the total exterior area of a three-dimensional shape like a rectangular box. Finding the surface area involves calculating all six faces and summing them up, which is done by:\[ A = 2(LW + LH + WH) \]If we have a requirement for volume but need to minimize the surface area, as in packaging cost considerations, it's vital to compute this carefully. For example, with the original box:\[ A = 2(6 \times 14 + 6 \times 20 + 14 \times 20) = 968 \text{ square inches} \]In the problem context, we seek boxes that maintain the same volume but with a lower surface area. This adjustment potentially results in less material used for manufacturing and can reduce costs.
Prealgebra Problem Solving
Prealgebra focuses on laying the groundwork for algebra, involving basic arithmetic and the understanding of fundamental concepts like geometry and measurement.
- **Identify the Problem:** Understand the specifications and requirements, such as achieving a certain volume while minimizing surface area.
- **Calculate Precisely:** Perform arithmetic on dimensions to derive volume and surface area, prioritizing accuracy in multiplication and addition.
- **Compare Results:** Evaluate options by comparing volumes and their corresponding surface areas to find what meets or fails the requirements.
