Math Problem Statement
A rectangular flower garden with an area of
494494
msquared2
is surrounded by a grass border 1 m wide on two sides and a 2 m wide on the other two sides as shown in the figure. What dimensions of the garden minimize the combined area of the garden and borders?
Question content area bottom
Part 1
The shorter edge (rectangular height) of the garden that will minimize the total area is
7 StartRoot 2 EndRoot72
m.
(Type an exact answer, using radicals as needed.)
Solution
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Math Problem Analysis
Mathematical Concepts
Optimization
Derivatives
Algebra
Formulas
Area of rectangle = length × width
Total area (garden + borders) = (x + 2)(y + 4)
y = 494494 / x
Minimization using derivative: dA/dx = -989988/x² + 4
Theorems
Critical Points Theorem
Optimization in Calculus
Suitable Grade Level
Grades 11-12
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