Tube A rectangular piece of cardboard, whose area is 216...

Name: Problem 86, Chapter 8: College Algebra
Uploaded: 2022-02-27T15:32:28-08:00
Duration: 6 min 12 s
Channel: Stephanie Carter
Description: Tube A rectangular piece of cardboard, whose area is 216 square centimeters, is made into a cylindrical tube by joining together two sides of the rectangle. See the figure. If the tube is to have a volume of 224 cubic centimeters, what size cardboard should you start with? CAN'T COPY THE IMAGE

Tube A rectangular piece of cardboard, whose area is 216 square centimeters, is made into a cylindrical tube by joining together two sides of the rectangle. See the figure. If the tube is to have a volume of 224 cubic centimeters, what size cardboard should you start with? CAN'T COPY THE IMAGE

Key Concepts

Cylinder Volume Formula

This concept involves understanding that the volume of a cylinder is given by the formula V = ?r²h, where r is the radius of the base and h is the height of the cylinder. It serves as a fundamental formula in solid geometry, allowing one to relate the dimensions of a cylindrical object to its volume, which is particularly useful when other constraints are present.

Correspondence between Rectangular and Cylindrical Dimensions

When a rectangle is rolled to form a cylindrical tube, one of its dimensions becomes the height of the cylinder, while the other forms the circumference of the base (with circumference represented by C = 2?r). This conversion is key to setting up relationships between the original two-dimensional shape and the resulting three-dimensional object.

Area Constraints in Geometric Design

The fixed area of the original piece of cardboard acts as a constraint on the dimensions of the rectangle. This concept is important in many geometric design problems as it allows the establishment of an equation that relates the length and width of a rectangle, serving as a boundary condition in the problem.

System of Equations from Geometric Constraints

By combining the equations derived from the cylinder’s volume formula and the area of the rectangular cardboard, one must set up and solve a system of equations. This process is crucial in solving problems where multiple conditions must be simultaneously satisfied, and it highlights the interplay between algebra and geometry.

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