I finally got a job. (Part 2)

Calculus Level 2

I am a carpenter who is building a rectangular room with a fixed perimeter of 180 ft. What is the area of the largest room that can be built?

Answer is in feet-squared, but you just write the number.


The answer is 2025.

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Lew Sterling Jr by Lew Sterling Jr , 24, USA

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2 solutions

Since both the dimensions are positive numbers, so we can apply the AM-GM inequality. Let length = a a and breadth = b b

P e r i m e t e r = 180 f t . 2 ( a + b ) = 180 a + b = 90 Perimeter=180ft.\\ 2(a+b)=180\\ a+b=90

Using AM-GM inequality,

a + b 2 ≥ ( a b 1 / 2 ) B u t , a + b = 90 90 2 ≥ ( a b 1 / 2 ) 45 ≥ ( a b 1 / 2 ) S q u a r i n g , 2025 ≥ a b \frac { a+b }{ 2 } \ge ({ ab }^{ 1/2 })\\ But,\quad a+b=90\\ \frac { 90 }{ 2 } \ge ({ ab }^{ 1/2 })\\ 45\ge ({ ab }^{ 1/2 })\\ Squaring,\\ 2025\ge ab

But a b ab is the are of the room whose maximum value = 2025

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Ramiel To-ong
Jul 28, 2015

To have a largest area it must be square in form thus; 180 = 4x x=45 A = 45^2 = 2025

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