Mathematics
Grade 10
15 min
Minimum and maximum area and volume
Minimum and maximum area and volume
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1
Introduction & Learning Objectives
Learning Objectives
Determine the dimensions of a rectangle with a maximum area for a given perimeter.
Determine the dimensions of a rectangle with a minimum perimeter for a given area.
Determine the dimensions of a rectangular prism with a maximum volume for a given surface area.
Determine the dimensions of a rectangular prism with a minimum surface area for a given volume.
Solve word problems involving optimization of area and volume.
Analyze and solve non-standard optimization problems, such as a three-sided enclosure.
How can you build the largest possible rectangular pen for your new puppy with a limited amount of fencing? 🐶
This lesson explores optimization in measurement, which is all about getting the 'most' for a given 'amount'. You will learn...
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Key Concepts & Vocabulary
TermDefinitionExample
OptimizationThe process of finding the best possible value (the maximum or minimum) of a quantity given certain limitations or constraints.Finding the dimensions of a rectangular garden that give the largest possible area using exactly 40 meters of fence.
ConstraintA limitation or condition that must be satisfied in an optimization problem.In the garden problem, the constraint is that the perimeter must be exactly 40 meters.
PerimeterThe total distance around the outside of a two-dimensional shape. For a rectangle, P = 2(l + w).A rectangle with sides 5 cm and 10 cm has a perimeter of 2(5 + 10) = 30 cm.
AreaThe amount of space inside a two-dimensional shape. For a rectangle, A = l × w.A rectangle with sides 5 cm and 10 cm has an area of 5 × 10 = 50 cm².
Surface AreaTh...
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Core Formulas
Maximum Area for a Fixed Perimeter (Rectangle)
For a given perimeter P, a square with side length `s = P/4` provides the maximum possible area.
Use this when you are given a fixed amount of 'boundary' material (like a fence) and want to enclose the largest possible rectangular area.
Minimum Perimeter for a Fixed Area (Rectangle)
For a given area A, a square with side length `s = \sqrt{A}` requires the minimum possible perimeter.
Use this when you need to enclose a specific rectangular area and want to use the least amount of 'boundary' material.
Maximum Volume for a Fixed Surface Area (Rectangular Prism)
For a given surface area SA, a cube with side length `s = \sqrt{SA/6}` provides the maximum possible volume.
Use this when you have a fixed amoun...
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Challenging
A farmer has 100 m of fencing to build a rectangular pen against a barn wall that is only 40 m long. The side of the pen parallel to the barn cannot be longer than the barn. What is the maximum area of the pen?
A.1250 m²
B.1200 m²
C.1000 m²
D.800 m²
Challenging
A square piece of land has an area of 400 m². It is fenced with the minimum possible perimeter. This same amount of fencing is then used to build a three-sided rectangular pen against a long wall. What is the maximum area of this new pen?
A.400 m²
B.800 m²
C.600 m²
D.1600 m²
Challenging
A box with a volume of 64 cm³ is to be constructed. The most efficient shape is a cube. If the dimensions were instead 2 cm x 4 cm x 8 cm, by how much does the surface area increase compared to the optimal cube?
A.The surface area is the same.
B.The surface area increases by 16 cm².
C.The surface area increases by 96 cm².
D.The surface area increases by 112 cm².
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