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