Math Problem Statement
a rectangular box has a volume of 144 cubic inches, a surface area of 192 sq inches, and a height 1in greater than its width. what is the distance from one corner of the box to the diagonally opposite corner?
Solution
Given:
- The volume of the rectangular box is 144 cubic inches.
- The surface area of the rectangular box is 192 square inches.
- The height is 1 inch greater than the width .
We are to find the distance from one corner of the box to the diagonally opposite corner.
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Let's denote:
- Width
- Height
- Length
-
Using the volume formula for a rectangular box: Substituting :
-
Using the surface area formula for a rectangular box: Substituting : Simplifying:
-
Solving for in terms of : From : Substituting into the surface area equation: Simplifying:
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Solve for by simplifying and solving the quadratic equation derived from the volume and surface area constraints. After simplification and solving, we get:
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Calculate the diagonal distance using the Pythagorean theorem in 3D:
Therefore, the distance from one corner of the box to the diagonally opposite corner is:
Would you like any details or have any questions?
Here are 8 related questions for further exploration:
- How would you find the volume of a rectangular prism if given different dimensions?
- Can the Pythagorean theorem be applied in 3D for any rectangular box?
- How do you derive the formula for the surface area of a rectangular prism?
- How does changing one dimension of the box affect its volume and surface area?
- What other shapes can you use the Pythagorean theorem in 3D for?
- How do you solve a quadratic equation by factoring?
- What is the significance of the diagonal in the context of a rectangular prism?
- How do you calculate the surface area if one dimension is unknown?
Tip: When dealing with 3D problems, visualizing the shape and drawing diagrams can significantly aid in understanding and solving the problem effectively.
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Math Problem Analysis
Mathematical Concepts
Geometry
Pythagorean Theorem
Quadratic Equations
Formulas
Volume of a rectangular box
Surface area of a rectangular box
Pythagorean theorem
Theorems
Pythagorean theorem
Suitable Grade Level
Grades 7-9
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