Designs that meet geometric constraints

Construct and compare geometric designs under stated constraints

Problem types

Modeling with geometry
Systems that model two constraints
Perimeter and area from coordinates · Useful idea
Designing a figure on a coordinate plane needs its vertices to give the required lengths and areas.
Volumes of cylinders, cones, pyramids, spheres · Useful idea
A design with a required volume leads to a constraint on its dimensions, such as the volume of a container.
The minimum of an upward-opening quadratic · Useful idea
The greatest or least value of a quadratic gives the best design, such as the largest fenced area.
Area and perimeter problems · Useful idea
A fixed amount of fence gives a rectangle whose area is found from one side length, a design constraint with a greatest area.

Quadratic, rational, and exponential constraints · Extend the idea
The area of a three-sided fenced garden, width times (fence minus two widths), is a quadratic constraint model with a greatest value.
Comparing uncertain strategies · See it applied
Choosing the best design among options compares their values, as a decision strategy compares expected values.
End behavior of a polynomial · See it applied
The volume of a box cut from a sheet is a cubic in the cut size, whose graph shows the greatest volume.

Modeling density · Related idea
Mass is density times volume, and the cost of painting a wall is a price per square meter times the painted area. Both multiply a rate per unit by a measured amount, which is how a design is priced.

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